On Feebly Compact Paratopological Groups 11

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1. Preliminaries In this paper the word “space” means “topological space”.

1.1. Topologized groups. A pair (G, τ) consisting of a group G and a topology τ on G is called a topologized group. A topologized group (G, τ) is called • a semitopological group if the topology τ is shift-continuous, that is for any a, b ∈ G the two-sided shift sa,b : G → G, sa,b : x 7→ axb, is continuous; • a quasitopological group if the topology τ is shift-continuous and the inversion G → G, x 7→ x−1, is continuous; • a paratopological group if τ is a semigroup topology, which means that the multiplication G × G → G, (x,y) 7→ xy, is continuous; • a topological group if τ is a group topology which means that τ is a semigroup topology with continuous inversion G → G, x 7→ x−1. It follows that a topologized group is a topological group if and only if it is both a paratopological group and a quasitopological group. For a shift-invariant topology τ on a group G by τ ♯ we denote the shift-invariant topology on G whose neighborhood base at the identity e of G consists of the sets U ∩ U −1 where e ∈ U ∈ τ. It follows that (G, τ ♯) is a quasitopological group, which is topologically isomorphic to the subgroup {(x,x−1) : x ∈ G} of G×G. If(G, τ) is a paratopological group, then (G, τ ♯) is a topological group, called the group coreﬂexion of (G, τ). In this paper we shall be mainly interested in paratopological groups. A classical example of a paratopological group failing to be a topological group is the Sorgenfrey line, that is the real line endowed with the Sorgenfrey topology (generated by the base consisting of half-intervals [a, b), a < b). For a semitopological group (G, τ) by τe = {U ∈ τ : e ∈ U} we denote the family of τ-open neighborhoods of the identity e of the group. Whereas the investigation of topological groups already is a fundamental branch of topological algebra (see, for instance, [68], [29], and [7]), semitopological groups are not so well- studied and have more variable structure. Basic properties of semitopological or paratopological groups compared with the properties of topological groups are described in the book [7] by Arhangel’skii and Tkachenko, in the PhD thesis of Ravsky [71], papers [69], [70], and in the survey [92] by Tkachenko.

1.2. Separation axioms. These axioms describe specific structural properties of spaces. Basic separation axioms and relations between them are considered in [36, Section 1.5]. For more specific cases and topics, also related to semitopological and paratopological groups, see [70, Section 1], [15], [92, Section 2], [94], [54]. All spaces considered in the present paper are not supposed to satisfy any of the separation axioms, if otherwise is not stated. For a τ subset A of a space (X,τ) by A and intτ A we denote the closure and interior of A in X, respectively. If the topology τ is clear from the context, then we write A and intA instead of τ A and intτ (A), respectively. Also by τ↾A we denote the subspace topology {U ∩ A : U ∈ τ} on A, inherited from the topology τ of X. Now we recall the definition of some separation axioms. A space X is

• T0 if for any distinct points x,y ∈ X there exists an open set U ⊆ X, which contains exactly one of the points x, y; • T1 if for any distinct points x,y ∈ X there exists a open set U ⊆ X such that x ∈ U and y∈ / U; • T2 or Hausdorﬀ if any distinct points x,y ∈ X have disjoint neighborhoods in X; ONFEEBLYCOMPACTPARATOPOLOGICALGROUPS 3

• regular if any neighborhood of any point contains a closed neighborhood of the point; • T3 if it is regular T0; • quasiregular if any nonempty open subset of X contains the closure of some nonempty open subset of X; • semiregular if X has a base consisting of regular open sets, that is sets U = int U; • functionally Hausdorff if for any distinct points x,y ∈ X there exists a continuous function f : X → R such that f(x) 6= f(y); • completely regular if for any closed set F ⊆ X and point x ∈ X \ F there exists a continuous function f : X → R such that f(x)=0 and f(F ) ⊆ {1}; • T 1 or Tychonoff if it is completely regular and T . 3 2 0 Remark that each regular space is quasiregular and semiregular. If a space (X,τ) satisfies a separation axiom T , we shall say that a topology τ is T . For instance, if (X,τ) is regular then τ is a regular topology. Given a space (X,τ), Stone [85] and Katˇetov [49] considered the topology τsr on X generated by the base consisting of all regular open sets of the space (X,τ). The topology τsr is called the semiregularization of the topology τ and the space (X,τsr) is called the semiregularization of the space (X,τ). It is easy to see that the semiregularization of a Hausdorff space is Hausdorff. 1.3. Separation axioms in paratopological groups. It is well-known that each topological group is completely regular. On the other hand, simple examples show that none of the implications T0 ⇒ T1 ⇒ T2 ⇒ T3 holds for paratopological groups (see [69, Examples 1.6-1.8] and page 5 in any of papers [70] or [92]) and there are only a few backwards implications between different separation axioms, see [70, Section 1] or [92, Section 2]. On the other hand, the authors proved in [15] that every semiregular paratopological group is completely regular and every Hausdorff paratopological group is functionally Hausdorff. The proof of the following important fact can be found in [70, Ex. 1.9], [71, p. 31] or [71, p. 28].

Lemma 1.1. For any (Hausdorff) paratopological group (G, τ), its semiregularization (G, τsr) is a (Hausdorff) regular paratopological group. Finally, let us mention that the paper [94] discusses some functors related to separation axioms in paratopological groups. 1.4. Compact-like spaces. Different classes of compact-like spaces and relations between them provide a well-known investigation topic in General Topology, see, for instance, basic [36, Chap. 3] and general [98], [84], [33], [64], [61] works. The inclusion relations between the classes are often visually represented by arrow diagrams, see, [64, Diag. 3 at p.17], [30, Diag. 1 at p. 58] (for completely regular spaces), [84, Diag. 3.6 at p. 611], and [44, Diag. at p. 3]. For recent results on embeddings of spaces into various Hausdorff compact-like spaces, see [10], [11], [12]. Let us recall the definitions of compact-like spaces which will appear in this paper. A space X is called • sequentially compact if each sequence in X contains a convergent subsequence; • ω-bounded if each countable subset of X has compact closure; • totally countably compact if each sequence of X contains a subsequence with compact closure; • countably compact at a subset A of X, if each infinite subset B of A has an accumulation point x ∈ X (the latter means that each neighborhood of x contains infinitely many points of the set B); 4 TARAS BANAKH AND ALEX RAVSKY

• countably compact if X is countably compact at itself, or, equivalently, if each countable open cover of X has a finite subcover; • sequentially pracompact if X contains a dense set D such that each sequence of points of the set D has a convergent subsequence; • countably pracompact if X is countably compact at a dense subset of X; • feebly compact if each locally finite family of open sets of the space X is finite; • sequentially feebly compact if for every sequence (Un)n∈ω of nonempty open sets of X there exists a point x ∈ X and an infinite set I ⊆ ω such that for every neighborhood Ox ⊆ X of x the set {n ∈ I : Ox ∩ Un 6= ∅} is finite; • pseudocompact if X is Tychonoff and each continuous real-valued function on X is bounded; • Lindelöf if each open cover of X has a countable subcover; • H-compact if any open cover U of X contains a finite subfamily V ⊆U such that X = SV ∈V V ; • H-closed if X is H-compact and Hausdorff; • countably o-compact if each decreasing sequence (Un)n∈ω of nonempty open sets in X has nonempty intersection Tn∈ω Un. A sequence {Un : n ∈ ω} of subsets of a space X is decreasing if Un ⊇ Un+1 for every n ∈ ω. These notions relate as follows: • Each compact space is ω-bounded and H-compact. • Each ω-bounded space is totally countably compact. • Each totally countably compact space is countably compact. • Each sequentially compact space is countably compact and sequentially pracompact. • Each countably compact space is countably pracompact. • Each sequentially pracompact space is countably pracompact. • Each countably pracompact space is feebly compact. • Each countably o-compact space is feebly compact. • Each Hausdorff countably o-compact space is finite. • A space is compact if and only if it is countably compact and Lindelöf. • A space is pseudocompact if and only if it is Tychonoff and feebly compact, see [36, Theorem 3.10.22]. • A space is H-compact if and only if its semiregularization is H-compact. • A Hausdorff space X is H-compact if and only if X is closed in any Hausdorff space containing X as a closed subspace, see [36, 3.12.5].

1.5. Compact-like semitopological groups. A semitopological group G is left (resp. right) precompact if for each neighborhood U of the identity of G, there exists a ﬁnite subset F of G such that FU = G (resp. UF = G). Proposition 2.1 from [70] implies that a paratopological group is left precompact iﬀ it is right precompact. So we will call left precompact paratopo- −1 logical groups precompact. A semitopological group G is 2-pseudocompact if Tn∈ω Un 6= ∅ for any decreasing sequence {Un : n ∈ ω} of nonempty open subsets of G. It is easy to see that a semitopological group is 2-pseudocompact if it is countably compact or countably o-compact. In Proposition 3.13 we shall prove that each 2-pseudocompact paratopological group is feebly compact. A semitopological group G is left ω-precompact if for each neighborhood U of the identity of G there exists a countable subset F of G such that FU = G. A semitopological group G ONFEEBLYCOMPACTPARATOPOLOGICALGROUPS 5

is saturated if for each nonempty open subset U of G there exists a nonempty open subset V of G such that V −1 ⊆ U. By [70, Proposition 3.1] each precompact paratopological group is saturated and by [70, Proposition 1.7] each saturated paratopological group is quasiregular. Recent investigations in papers [8], [55], and [56] are using the following notions. A topologized group G is called pseudobounded provided for each neighborhood U of the identity of the group G there exists a natural number n such that U n = G. A topologized group G is called ω- n pseudobounded provided G = Sn∈N U for any neighborhood U of the identity in G. Clearly, every pseudobounded topologized group is ω-pseudobounded. However, the additive group (R, +) endowed with the standard topology is an ω-pseudobounded topological group which is not pseudobounded. The Sorgenfrey line is not even ω-pseudobounded.

1.6. Automatic continuity of operations in semitopological groups. It turns out that if the space of a semitopological (resp. paratopological) group satisfies suitable conditions (sometimes with some conditions imposed on the group), then the multiplication (resp. inversion) in the group is continuous, i.e., the group is paratopological (resp. topological). Finding such conditions is one of main branches of the theory of paratopological groups. It turns out that automatic continuity essentially depends on compact-like properties and separation axioms of the space of a semitopological group. An interested reader can find known results and references on this subject in Section 5.1 of [71] and Section 3 of the survey [92] (both for semitopological and paratopological groups), in Introduction of [1] and in the present paper (for paratopological groups). We briefly recall the history of the topic. In 1936 Montgomery [65] showed that every completely metrizable paratopological group is a topological group. In 1953 Wallace [99] asked whether every regular locally compact semitopological group a topological group. In 1957 Ellis [34], [35] obtained a positive answer to the Wallace question (later the second author of the present paper showed that the regularity condition can be relaxed, see Proposition 5.5 in [71] or its counterpart in English in [74]). In 1960 Zelazko˙ used Montgomery’s result and showed that each completely metrizable semitopological group is a topological group. Since locally compact spaces and completely metrizable spaces are Cech-ˇ complete (recall that Cech-completeˇ spaces are Gδ-subspaces of compact Hausdorff spaces), this motivated Pfister [67] in 1985 to ask whether each Cech-completeˇ semitopological group is a topological group. In 1996 Bouziad [18] and Reznichenko [77], as far as the authors know, independently answered affirmatively the Pfister question. To do this, it was sufficient to show that each Cech-completeˇ semitopological group is a paratopological group since ear- lier, Brand [20] had proved that every Cech-completeˇ paratopological group is a topological group. Brand’s proof was later improved and simplified in [67]. If G is a paratopological group which is a T1 space and G × G is countably compact (in particular, if G is sequentially compact) then G is a topological group, see [75]. On the other hand, in the present paper we show that we cannot weaken T1 to T0 here, constructing in Example 5.27 a sequentially compact T0 paratopological group which is not a topological group. Also we cannot weaken the countable compactness of G × G to that of G because under additional axiomatic as- sumptions there exists a countably compact paratopological group which is not a topological group, see Example 3.22. Also by Example 3.30, there exists a Hausdorff second-countable sequentially pracompact paratopological group G (so by [44, Proposition 2.1] each its power is sequentially pracompact) such that G is not a topological group. Remark that examples of compact-like paratopological groups that fail to be topological groups can be constructed by means of so-called cone topologies, see Section 5. On the other hand, by Proposition 3.15, each feebly compact quasiregular paratopological group is a topological group. In particular, 6 TARAS BANAKH AND ALEX RAVSKY each pseudocompact paratopological group is a topological group, see also [6, Theorem 1.7] and [4, Theorem 2.1].

2. The Ra˘ıkov completion of a paratopological group

In this section we describe a canonical construction of extension of a T0 paratopological group, which coincides with the Ra˘ıkov completion if the paratopological group is a topological group. Let us recall that a Hausdorff topological group G is Ra˘ıkov-complete if it is complete in the two-sided uniformity. This uniformity is generated by the family of the entourages {(x,y) ∈ G × G : x ∈ Uy ∩ yU}, where U runs over neighborhoods of the identity in G. It is known that each Hausdorff topological group G is a dense subgroup of a Ra˘ıkov complete topological group G˘, which is unique up to a topological isomorphism. This unique topological group G˘ is called the Ra˘ıkov completion of G, see [7, §3.6] for more details. In this section we extend the construction of Ra˘ıkov completion to all T0 paratopological groups. A paratopological group (G, τ) is defined to be ♯-complete if its group coreflexion (G, τ ♯) is Hausdorff and Ra˘ıkov complete. Let us recall that a neighborhood base of the topology τ ♯ at the identity e of G consists of the intersections U ∩U −1 where e ∈ U ∈ τ. The Hausdorff property of the topology τ ♯ implies that the topology τ is T0. Therefore, each ♯-complete paratopological group satisfies the separation axiom T0. In this section we shall construct an embedding of any T0 paratopological group G into a ♯-complete paratopological group G˘, called the Ra˘ıkov completion of the paratopological group G. The construction of the Ra˘ıkov completion exploits a construction of extension of a τ-regular semigroup topology σ on a subgroup H of a semitopological group (G, τ) to a semigroup topology στ on G. This construction will be described and analyzed in Subsection 2.1. Its applications to Ra˘ıkov completions will be given in Subsection 2.2. We shall need the following property of semiregularizations of paratopological groups. Lemma 2.1. Let (G, τ) be a semitopological group, H be a dense subgroup of (G, τ) and σ be a regular semigroup topology on H such that σ ⊆ τ↾H. Then σ ⊆ τsr↾H. In particular, if τ↾H = σ then τsr↾H = σ. Proof. Let U ⊆ H be an arbitrary neighborhood of the identity in the topology σ. By σ the regularity of σ, there exists an open neighborhood V ∈ σe such that V ⊆ U. Since σ ⊆ τ↾H, there exists an open neighborhood W ∈ τe such that W ∩ H ⊆ V . Taking into τ τ τ account that the subgroup H is dense in (G, τ), we see that W = W ∩ H ⊆ V . Then τ τ σ W ∩ H ⊆ V ∩ H ⊆ V ⊆ U. 2.1. Extending topologies from subgroups to groups. Let H be a subgroup of a semitopological group (G, τ). For any semigroup topology σ on H, consider the topology στ on G consisting of the subsets U ⊆ G such that for any x ∈ U there exists a neighborhood V ∈ σ τ of the identity such that xV ⊆ U. τ Lemma 2.2. For every neighborhood U ∈ σ of e and any x ∈ G the set xU is a neighborhood of x in the topology στ .

Proof. Since σ is a semigroup topology, we can choose a sequence (Un)n∈ω of neighborhoods of the identity in (H,σ) such that U0U0 ⊆ U and UnUn ⊆ Un−1 for every n ∈ N. The last condition implies that

U0 · · · Un ⊆ U0 · · · Un−1Un−1 ⊆ U0 · · · Un−2Un−2 ⊆···⊂ U0U0 ⊆ U ONFEEBLYCOMPACTPARATOPOLOGICALGROUPS 7

for each n ∈ ω. Taking into account that (G, τ) is a semitopological group, we conclude that

U0 · · · Un ⊆ U0 · · · Un ⊆ U where the closure is taken in the topology τ. Then the union W := Sn∈ω U0 · · · Un is a subset of U and xW ⊆ xU. We claim that the set xW belongs to the topology στ . Indeed, for any y ∈ xW , we can ﬁnd m ∈ ω such that y ∈ xU0 · · · Um and observe that yUm+1 ⊆ τ xU0 · · · Um Um+1 ⊆ xW . Therefore, xU ⊇ xW ∈ σ is a neighborhood of x in the topology στ .

We shall prove thatσ ¯τ is a semigroup topology on G if the topology σ is τ-balanced in the sense that for any neighborhood U ∈ σe and any x ∈ G there exists a neighborhood V ∈ σe τ such that xV x−1 ⊆ U . A topology σ on H is defined to be τ-regular if for any neighborhood τ U ∈ σe, there exists a neighborhood V ∈ σe such that H ∩ V ⊆ U. Observe that a regular topology σ on H is τ-regular if σ ⊆ τ↾H. Lemma 2.3. Let H be a subgroup of a semitopological group (G, τ) and σ be a τ-balanced semigroup topology on H. Then (1) στ is a semigroup topology on G such that στ ↾H ⊆ σ; (2) If σ ⊆ τ↾H and the set H is dense in (G, τ), then στ ⊆ τ; τ (3) If σ ⊆ τ↾H, then σsr ⊆ σ ↾H; τ (4) If σ ⊇ τ↾H and τ is a semigroup topology, then τsr ⊆ σ ; (5) If the topology σ is τ-regular, then σ = στ ↾H; τ (6) If σ ⊆ τ↾H and the space (H,σ) is regular, then σ = σsr = σ ↾H. (7) If σ ⊆ τ↾H, the space (H,σ) is regular, and H is dense in (G, τ), then σ = σsr = τ τ σ ↾H = (σ )sr ↾H. Proof. 1. The definition of the topology στ ensures that στ ↾H ⊆ σ. Next, we show that στ τ is a semigroup topology. Given any points x,y ∈ G and a neighborhood Oxy ∈ σ¯ of xy, use Lemma 2.2 to find a neighborhood U ∈ σ of e such that xyU ⊆ Oxy, where the closure is taken in the topology τ. Since σ is a semigroup topology, there exists a neighborhood V ∈ σ of e such that VV ⊆ U. By the τ-balanced property of the topology σ, there exists a −1 neighborhood W ∈ σe such that W ⊆ V and y Wy ⊆ V . Since (G, τ) is a semitopological group, y−1Wy ⊆ V . By Lemma 2.2, the sets xW and yW are neighborhoods of the points x,y, respectively in the topology στ . Finally, observe that

−1 xWyW = xy(y Wy)W ⊆ xyV · W ⊆ xyVW ⊆ xyVV ⊆ xyU ⊆ Oxy, witnessing that στ is a semigroup topology on G. 2. Assume that σ ⊆ τ↾H and H is dense τ τ in (G, τ). To show that σ ⊆ τ, fix any neighborhood W ∈ σe of the identity and using the τ τ definition of the topology σ , find a neighborhood U ∈ σe such that U ⊆ W . Since σ ⊆ τ↾H, there exists a neighborhood V ∈ τe such that V ∩ H ⊆ U. The density of H in (G, τ) implies that τ τ τ V ⊆ V = V ∩ H ⊆ U ⊆ W, τ witnessing that σ ⊆ τ. 3. Let Usr be an arbitrary neighborhood of the identity in the semireg-

ularization σsr of the topology σ. By Lemma 1.1, the topology σsr is regular. Consequently, σsr there exists an open neighborhood Vsr ∈ σsr of the identity such that Vsr ⊆ Usr. By the 8 TARAS BANAKH AND ALEX RAVSKY deﬁnition of the topology σsr, there exists a regular open set V ∈ σe such that V ⊆ Vsr. By τ Lemma 2.2, the set V is a neighborhood of the identity in the topology στ . Since σ ⊆ τ↾H,

τ τ τ↾H σ σsr V ∩ H ⊆ Vsr ∩ H = Vsr ⊆ Vsr ⊆ Vsr ⊆ Usr. τ 4. Assuming that τ↾H ⊆ σ and τ is a semigroup topology, we shall show that τsr ⊆ σ . By Lemma 1.1, the topology τsr is regular. Given any neighborhood Usr ∈ τsr of the identity, use the regularity of the topology τsr and find a neighborhood Vsr ∈ τsr of the identity such τsr that Vsr ⊆ Usr. By definition of the topology τsr, there exists a regular open neighborhood V ∈ τ such that V ⊆ Vsr. Since τ↾H ⊆ σ, the set V ∩ H is a neighborhood of the identity in τ (H,σ). By Lemma 2.2, the closure V ∩ H contains a neighborhood W ∈ στ of the identity. τ τ τsr Then W ⊆ V ⊆ Vsr ⊆ Vsr ⊆ Usr. 5. The statement follows from Lemma 2.2. 6. If σ ⊆ τ↾H and the space (H,σ) is regular, then the topology σ is τ-regular and then σ =σ ¯τ ↾H by the preceding statement. The regularity of the topology σ ensures that σ = σsr. 7. Finally assume that σ ⊆ τ↾H, the space (H,σ) is regular, and H is dense in (G, τ). By the preceding τ statement, σ = σsr = σ ↾H. By the statement (2), the density of H in (G, τ) implies the τ τ density of H in (G, σ ). Now Lemma 2.1 implies that (σ )sr ↾H = σ. Now we establish sufficient conditions of the τ-balancedness and τ-regularity of the topology σ. Lemma 2.4. Let H be a subgroup of a paratopological group (G, τ) and σ be a semigroup topology on H such that σ ⊆ τ ♯↾H and H is dense in (G, τ ♯). Then the topology σ is τ- balanced. Proof. The inclusion σ ⊆ τ ♯↾H and the invariance of the topology τ ♯ under the inversion ♯ ♯ imply that σ ⊆ τ ↾H. Given any neighborhood U ∈ σe and point x ∈ G, we need to find a −1 τ neighborhood V ∈ σe such that xV x ⊆ U . Since σ is a semigroup topology, there exists 3 −1 a neighborhood W ∈ σe such that W ⊆ U. The set W ∩ W belongs to the topology σ♯ ⊆ τ ♯↾H. Taking into account that the subgroup H is dense in (G, τ ♯), we see that the τ ♯ τ ♯-closure W ∩ W −1 of the set W ∩ W −1 has nonempty interior in (G, τ ♯) and hence has τ ♯ τ a common point y with the dense set xH. It follows that y ∈ W ∩ W −1 ⊆ W and τ ♯ τ ♯ −1 −1 −1 −1 τ y ∈ W ∩ W = W ∩ W ⊆ W . Since (H,σ) is a semitopological group and x−1y ∈ H, the set V := x−1yWy−1x is a neighborhood of the identity in (G, σ). Moreover, τ τ τ τ xV x−1 = xx−1yWy−1xx−1 = yWy−1 ⊆ W W W ⊆ WWW ⊆ U . Lemma 2.5. Let H be a subgroup of a semitopological group (G, τ). A semigroup topology σ on H is τ-regular if σ−1 ⊆ τ↾H, where σ−1 = {U −1 : U ∈ σ}. Proof. To prove that σ is τ-regular, fix any neighborhood U ∈ σ of the identity. Since σ is a semigroup topology, there exists a neighborhood V ∈ τe such that VV ⊆ U. Taking into account that σ−1 ⊆ τ↾H, conclude that −1 τ τ↾H σ −1 V ∩ H = V ⊆ V = \ VW = \ VW ⊆ VV ⊆ U, e∈W ∈σ−1 e∈W ∈σ witnessing that the topology σ is τ-regular. ONFEEBLYCOMPACTPARATOPOLOGICALGROUPS 9

2.2. The Ra˘ıkov completion of a paratopological group. Given a T0 paratopological group (G, τ), consider its group coreflexion G♯ = (G, τ ♯), which is a Hausdorff topological group. Let G˘♯ = (G,˘ τ˘♯) be the Ra˘ıkov completion of the topological group G♯. Observe that τ ⊆ τ ♯ =τ ˘♯↾G. By Lemmas 2.4 and 2.5, the topology τ isτ ˘♯-balanced andτ ˘♯-regular. ♯ Consequently, the semigroup topology τ τ˘ on G˘ is well-defined. To simplify notations, we ♯ shall denote the topology τ τ˘ byτ ˘. The paratopological group (G,˘ τ˘) will be called the Ra˘ıkov completion of the paratopological group (G, τ) and will be denoted by G˘. By Lemma 2.2, a neighborhood base at the identity in G˘ consists ofτ ˘♯-closures of neighborhoods of the identity in the paratopological group (G, τ). Lemma 2.3 implies the following properties of the Ra˘ıkov completions of paratopological groups.

Theorem 2.6. For a T0 paratopological group (G, τ) its Ra˘ıkov completion (G,˘ τ˘) has the following properties:

(1) τsr ⊆ τ =τ ˘↾G; (2)τ ˘ ⊆ τ˘♯ = (˘τ)♯; (3) the topology τ˘ satisﬁes the separation axiom T0; ♯ ♯ ♯ (4) If the space (G, τ) is regular, then τ =τ ˘↾G =τ ˘sr↾G, τ˘ = (˘τ) = (˘τsr) , and the topologies τ˘sr ⊆ τ˘ are Hausdorﬀ; (5) If (G, τ) is a topological group, then τ˘ =τ ˘♯.

Proof. 1. Lemma 2.5 implies that the topology τ isτ ˘♯-regular. By Lemma 2.3(5), theτ ˘♯- regularity of the topology τ implies the equality τ =τ ˘↾G. The inclusion τsr ⊆ τ follows from the definition of the semiregularization of the topology τ. 2. The inclusionτ ˘ ⊆ τ˘♯ follows from Lemma 2.3(2). Applying to this inclusion the operation of group coreflexion, we obtain (˘τ)♯ ⊆ (˘τ ♯)♯ =τ ˘♯. To show thatτ ˘♯ ⊆ (˘τ)♯, take any neighborhood U ∈ τ˘♯ of the identity. Using the regularity of the group topologyτ ˘♯, find a neighborhood W ∈ τ˘♯ of the identity such that W ⊆ U where the closure of W is taken in the topologyτ ˘♯. By the definition of the topology τ ♯ =τ ˘♯↾G, there exists a neighborhood V ∈ τ such that V ∩ V −1 ⊆ W ∩ G. Since τ =τ ˘↾G, there exists a set T ∈ τ˘ such that T ∩ G = V . Since T ∩ T −1 ∈ (˘τ)♯ ⊆ τ˘♯, the intersection T ∩ T −1 ∩ G is dense in T ∩ T −1. Consequently,

T ∩ T −1 ⊆ T ∩ T −1 ∩ G = V ∩ V −1 ⊆ W ⊆ U and hence U is the neighborhood of the identity in the topology (˘τ)♯. 3. To see that the topologyτ ˘ satisfies the separation axiom T0, take any distinct points x,y ∈ G˘. Using the T1 property of the group topologyτ ˘♯ = (˘τ)♯, find a neighborhood U ∈ τ˘ of the identity such that x−1y∈ / U ∩ U −1. If x−1y∈ / U, then xU ∈ τ˘ is a neighborhood of x that does not contain y. If x−1y ∈ U, then x−1y∈ / U −1 and then yU ∈ τ˘ is a neighborhood of y that does not contain x. 4. If the space (G, τ) is regular, then τ =τ ˘↾G =τ ˘sr↾G by Lemma 2.3(7). The inclusion ♯ ♯ ♯ ♯ ♯ ♯ τ˘sr ⊆ τ˘ ⊆ τ˘ established in the statement (2) implies (˘τsr) ⊆ (˘τ) ⊆ (˘τ ) =τ ˘ . To see ♯ ♯ ♯ thatτ ˘ ⊆ (˘τsr) , fix any neighborhood U ∈ τ˘ of the identity. By the regularity of the group topologyτ ˘♯, there exists a neighborhood W ∈ τ˘♯ of the identity such that W ⊆ U, where the closure in taken in the topological group G˘♯. Since W ∩ G ∈ τ ♯, there exists a neighborhood −1 V ∈ τ of the identity such that V ∩ V ⊆ W ∩ G ⊆ W . Since τ =τ ˘sr↾G, there exists a −1 −1 neighborhood B ∈ τ˘sr of e such that B ∩ G ⊆ V . Then B ∩ B ∩ G ⊆ V ∩ V ⊆ W . Since 10 TARAS BANAKH AND ALEX RAVSKY the set B ∩ B−1 is open in the topological group G˘♯, B ∩ B−1 ⊆ B ∩ B−1 = B ∩ B−1 ∩ G ⊆ W ⊆ U. −1 ♯ Taking into account that U ⊇ B ∩ B ∈ (˘τsr) , we conclude that U is a neighborhood of ♯ ♯ ♯ the identity in the topology (˘τsr) and henceτ ˘ ⊆ (˘τsr) . By analogy with the proof of the ♯ ♯ T0 property of the topologyτ ˘, we can prove that the equalityτ ˘ = (˘τsr) implies the T0 separation axiom for the topologyτ ˘sr. Being regular, the T0 topologyτ ˘sr is Hausdorff and ♯ ♯ so is the topologyτ ˘ ⊇ τ˘sr. 5. If (G, τ) is a topological group, then τ =τ ˘ ↾G andτ ˘ =τ ˘ by Lemma 2.3(2,4). Theorem 2.6(1–3) implies the following corollary.

Corollary 2.7. Every T0 paratopological group G is a dense subgroup of the ♯-complete paratopological group G˘. Theorem 2.6(4) and Lemma 1.1 imply the following corollary.

Corollary 2.8. Every T3 paratopological group G is a dense subgroup of the regular ♯-complete paratopological group G˘sr, which coincides with the semiregularization of the Ra˘ıkov completion G˘ of G.

The following simple example shows that separation axioms T1 and T2 are not inherited by the Ra˘ıkov completions of paratopological groups. Example 2.9. There exists a countable Abelian ﬁrst-countable Hausdorﬀ paratopological group G whose Ra˘ıkov completion G˘ does not satisfy the separation axiom T1. Moreover, the closure of the singleton {0} in G˘ is not compact. Proof. Take any countable dense subgroup G in the additive group R of real numbers. Take any non-zero real number c such that G ∩ (Z · c) = {0}. For every n ∈ ω consider the set −n Un = Sk∈ω{g ∈ G : |g − kc| < 2 }. Observe that Un+1 + Un+1 ⊆ Un for every n ∈ ω, which implies that the topology

τ = U ⊆ G : ∀x ∈ U ∃n ∈ ω (x + Un ⊆ U)} turns G into a first-countable paratopological group. The paratopological group (G, τ) is Hausdorff since for any distinct elements x,y ∈ G there exists n ∈ ω such that x − y∈ / −n Sk∈Z{g ∈ G : |g − kc| < 2 }, which implies that x + Un+1 is disjoint with y + Un+1. The topology τ ♯ of the group coreflexion of the paratopological group (G, τ) coincides with the Euclidean topology on G and the Ra˘ıkov completion of the topological group (G, τ ♯) can be identified with the real line. The definition of the topology τ ensures that for any n ∈ ω the closure of the set Un in the real line contains the set {kc : k ≥ 0}. Consequently, the closure of {0} in the Ra˘ıkov completion G˘ of (G, τ) contains the set {kc : k ≤ 0} and hence is not compact. Remark 2.10. After writing down the above construction of the Ra˘ıkov completion of a T0 paratopological group, we have discovered that essentially the same completion was elab- orated by Mar´ın and Romaguera [63] who observed that the bicompletion of the two-sided quasiuniformity of a paratopological group has a natural structure of a paratopological group that contains the group as a 2-dense subgroup. In contrast, our construction is more direct and does not involve quasiuniformities. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 11

3. When is a feebly compact paratopological group a topological group? The following lemma proved by Tkachenko [94, Theorem 3.1] allows us to reduce topological questions related to semitopological group to the case of T0-spaces. Lemma 3.1. Let G = (G, τ) be a semitopological group and H := (U ∩ U −1). Then TU∈τe H is a normal subgroup of G, the quotient space T0G := G/H satisﬁes the separation axiom −1 T0 and each open set U ⊆ G is equal to π (π(U)) where π : G → G/H is the quotient homomorphism.

The quotient group T0G := G/H is called the T0-reﬂexion of G. If G is a (para)topological group, then so is its T0-reﬂexion T0G, see [94]. Proposition 3.2. Each totally countably compact paratopological group is a topological group.

Proof. Let G = (G, τ) be such a group. Put B = T τe. The continuity of the multiplication ′ implies that B is a subsemigroup of the group G. Put B = {e} = {x ∈ G : ∀U ∈ τe (e ∈ xU)} = {x ∈ G : e ∈ xB} = B−1. Hence B′ is a closed subsemigroup of the group G. The total countable compactness of G implies that B′ = {e} compact. Being a compact topological semigroup, B′ contains a minimal closed right ideal I ⊂ B′. Since I is closed in G, for any element x ∈ I, the set x2I ⊆ B′ is closed in G and in B′. By the minimality of I, x2I = I ∋ x. Hence x−1 ∈ I and e ∈ I. Consequently I = B′ and xB′ = B′ for each element ′ ′ ′ −1 x ∈ B . Therefore B is a group and B = B. Since B = T τe, we see that g Bg ⊆ B for each g ∈ G. Hence B is a normal subgroup of the group G. Since B has the antidiscrete topology, B is a topological group. Since the normal subgroup B is closed in G, the quotient group G/B is a T1-space. Let π : G → G/B be the quotient homomorphism. Given any sequence (yn)n∈ω in the quotient group G/B, choose a sequence (xn)n∈ω in the group G such that π(xn)= yn for each n ∈ ω. Since G is totally countable compact, there is a subsequence A of {xn : n ∈ ω} such that the closure A is compact. Since the set A is closed, A = AB. Hence the closed compact set π(A) contains a subsequence of the sequence {yn : n ∈ ω}. Therefore G/B is a T1 totally countably compact paratopological group. By [1, Cor 2.3], G/B is a topological group. Hence both B and G/B are topological groups. Then by [72, 1.3] or [71, Proposition 5.3], G is a topological group too. Proposition 3.2 generalizes Corollary 2.3 in [1] and Lemma 5.4 in [71]. Lemma 3.3. [70, Proposition 1.7] Each saturated paratopological group is quasiregular. Let us recall [17] that a paratopological group G is topologically periodic if for each x ∈ G and a neighborhood U ⊆ G of the identity there is a number n ≥ 1 such that xn ∈ U. Reznichenko proved the following lemma, announced in [75]. Lemma 3.4. A paratopological group (G, τ) is a topological group if and only if (G, τ) is quasiregular and its semiregularization (G, τsr) is a topological group. Proof. The “only if” part follows trivially from the regularity of any topological group. To prove the “if” part, assume that (G, τ) is quasiregular and its semiregularization (G, τsr) is a topological group. To show that G is a topological group, ﬁx any neighborhood N ∈ τ of e. Since (G, τ) is a paratopological group, there exists a neighborhood U ∈ τ of e such that UU ⊆ N. Consider the regular open set W := int U ∈ τsr and observe that the open set V = S{T ∈ τ : T ⊆ U} is dense in U (by the quasiregularity of the topology τ) and belongs to the topology τsr, being the union of canonically open sets in the topology τ. Consequently, 12 TARAS BANAKH AND ALEX RAVSKY

−1 V ⊆ U ⊆ W and V = U = W . Since (G, τsr) is a topological group, the sets W and −1 −1 V are τsr-open. The density of the set V in W implies the density of the set V ∩ V in −1 −1 W ∩ W in the topological group (G, τsr). Since W ∩ W ∋ e is nonempty, the open set −1 −1 V ∩ V ∈ τsr ⊆ τ is not empty, too. Take any point x ∈ V ∩ V and observe that N −1 ⊇ (UU)−1 ⊇ V −1V −1 ⊇ x−1(V ∩ V −1) is a neighborhood of e in the topology τ.

The following lemma easily follows from the deﬁnitions.

Lemma 3.5. A space (X,τ) is feebly compact if and only if its semiregularization (X,τsr) is feebly compact. The following two lemmas were proved by Alas and Sanchis [1] in case of paratopological groups satisfying the separation axiom T0. But Lemma 3.1 allows us to remove the T0 assumption. Lemma 3.6. Each topologically periodic Baire paratopological group is saturated. Lemma 3.7. Each 2-pseudocompact paratopological group is a Baire space. Lemma 3.8. Each feebly compact topological group G is precompact. Proof. If G is not precompact, then there exists a neighborhood U ⊆ G of the identity and a sequence of points {xn}n∈ω ⊆ G such that xm ∈/ xnU for any n

The following lemma was first proved in the Ph.D. dissertation [71, 3.1] of the first author. Lemma 3.9. Every precompact paratopological group (G, τ) is saturated. Proof. Given any neighborhood U ∈ τ of the identity e, choose a neighborhood V ∈ τ of e such that VV ⊆ U. By the precompactness of (G, τ), there exists a finite set F ⊆ G such that G = FV . Since G = V −1F −1, for some x ∈ F −1 the set V −1x−1 is not nowhere dense in G. Then the closure V −1 contains some nonempty open set W ∈ τ. Now observe that

−1 −1 −1 −1 −1 W ⊆ V −1 = \ V T ⊆ V V ⊆ U , T ∈τe which means that the set U −1 has nonempty interior. Therefore, the paratopological group (G, τ) is saturated.

By analogy one can prove the following lemma, see [1, Theorem 2.5]. Lemma 3.10. Each Baire left ω-precompact paratopological group is saturated. The following lemma can be derived from [6, Theorem 1.7] and Lemma 3.1.

Lemma 3.11. Each paratopological group that is a dense Gδ-subset of a regular feebly compact space is a topological group. Lemma 3.12. Each regular 2-pseudocompact paratopological group is a topological group. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 13

Proof. Let (G, τ) be such a group. Suppose (G, τ) is not a topological group. Then there −1 −1 2 exists a neighborhood U ∈ τ of the unit e such that V 6⊆ U ⊆ (U ) for every V ∈ τe. By induction we can build a sequence {Ui}i∈ω ∈ τ of open neighborhoods of e such that U0 ⊆ U 2 and Ui+1 ⊆ Ui for each i ∈ ω. Then F =: Ti∈ω Ui is a subsemigroup of G. Since the group G is 2-pseudocompact, there is a point x ∈ G such that

−1 −1 −1 −1 2 −1 x ∈ \ (Ui\U ) ⊆ \ Ui ⊆ \(Ui ) ⊆ F . i∈ω i∈ω i∈ω

−1 −1 −1 −1 −1 Moreover, x ∈ (U0\U ) ⊆ (U0\U ) ⊆ U0 \U ⊆ G\U. Since x 6∈ U0 there is a neighborhood W ∈ τe such that W ⊆ U1 and Wx ∩ U0 = ∅. Let m,n ∈ ω and n>m ≥ 1. If n m m+1−n 2 ∅= 6 Wx ∩Wx then ∅= 6 Wx∩Wx ⊆ Wx∩U1F ⊆ Wx∩U1 ⊆ Wx∩U0 = ∅. Therefore i {Wx : i ≥ 1} is an inﬁnite family of pairwise disjoint sets. Choose a neighborhood V ∈ τe 2 i −1 such that V ⊆ W . Since G is 2-pseudocompact, there is a point y ∈ Tn≥1 (Si≥n Vx ) . There exist numbers m,n ∈ ω such that n>m ≥ 1 and yV ∩ (Vxm)−1 6= ∅= 6 yV ∩ (Vxn)−1. Then y−1 ∈ V 2xm ∩ V 2xn ⊆ Wxm ∩ Wxn. This contradiction proves that G is a topological group. Observe that a paratopological group is 2-pseudocompact if it is either countably compact or it is a feebly compact topological group. Proposition 3.13. Each 2-pseudocompact paratopological group is feebly compact.

Proof. Let G be such a group. The semiregularization Gsr of G is a regular 2-pseudocompact paratopological group. By Lemma 3.12, (G, τsr) is a topological group. The 2-pseudocompactness of the topological group Gsr implies the feeble compactness of Gsr. By Lemma 3.5, the paratopological group G is feebly compact. Proposition 3.13 cannot be reversed: in [81, Theorem 2] Tkachenko and Sanchis have constructed a Hausdorﬀ feebly compact Baire paratopological group which is not 2-pseudocompact, thus answering a question of the second author from an older version of this manuscript. Problem 3.14. Is each countably pracompact Baire paratopological group 2-pseudocompact? The answer to Problem 3.14 is positive if the paratopological group is quasiregular or saturated or topologically periodic or left ω-precompact. Proposition 3.15. For a feebly compact paratopological group G, the following conditions are equivalent: 1) G is quasiregular; 2) G is saturated; 3) G is topologically periodic and Baire; 4) G is left ω-precompact and Baire; 5) G is precompact; 6) G is a topological group. Proof. It is easy to prove that Condition 6 implies all other conditions (remark that a feebly compact topological group is 2-pseudocompact and hence Baire by Lemma 3.7). (1) ⇒ (6) By Lemma 1.1, the semiregularization Gsr of the paratopological group G is a regular feebly compact paratopological group. By Lemma 3.11, Gsr is a topological group. Then by Lemma 3.4, G is a topological group, too. The implication (2) ⇒ (1) follows from Lemma 3.3. The implications (3, 4, 5) ⇒ (2) follow from Lemmas 3.6, 3.10, and 3.9, respectively. 14 TARAS BANAKH AND ALEX RAVSKY

The implication (1) ⇒ (6) in Proposition 3.15 generalizes Proposition 2 in [75]; (4) ⇒ (6) generalizes Proposition 2.6 in [1]; the implication (3) ⇒ (6) answers Question C in [1] and generalizes Theorem 3 in [17]. A group has two basic operations: the multiplication and the inversion. The first operation is continuous on a paratopological group, but the second can be discontinuous. But there are continuous operations on a paratopological group besides the multiplication. For instance, a power. Now suppose that there exists a nonempty open subset U of a paratopological group G such that the inversion on the set U coincides with a continuous operation. Then the continuity of the inversion on U can be used to show that the group G is topological. Let us consider a trivial application of this idea. Let G be a paratopological group of bounded exponent. Then there exists a number n ∈ N such that the inversion on the group G coincides with the n-th power. Thus the group G is topological. Another applications are formulated in two next propositions. Let us recall that a group is periodic if each element of the group has finite order. Proposition 3.16. Each Baire periodic paratopological group is a topological group. ′ −1 ′ Proof. Let (G, τ) be such a group. Put B = T{U : U ∈ τe}. Then B is closed subset of G. Since B′ is a periodic semigroup, we see that B′ is a group such that B′ = (B′)−1 ⊆ U for n ′ any neighborhood U of the identity. For each n > 1 we put Gn = {x ∈ G : x ∈ B }. Since the space (G, τ) is Baire, there is a number n > 1 such that the set Gn has the nonempty interior in (G, τ). Therefore there are a point x ∈ Gn and a neighborhood U ∈ τe such that n ′ −1 ′ n−1 (xu) ∈ B for each u ∈ U. Then V ⊆ B (xV ) x for each subset V of U. Let U1 ∈ τ n ′ be an arbitrary neighborhood of the identity. Then x ∈ B U1. By the continuity of the n−1 ′ multiplication there is a neighborhood W ⊆ U of the identity such that (xW ) x ⊆ B U1. −1 ′ n−1 ′ ′ ′ ′ Then W ⊆ B (xW ) x ⊆ B B U1 = B U1. Taking into account that B ⊆ U1, we see that −1 ′ 2 W ⊆ B U1 ⊆ U1 . Proposition 3.17. Any Hausdorff feebly compact periodic paratopological group is a topological group.

Proof. Let (G, τ) be such a group. By Lemma 1.1, its semiregularization (G, τsr) is a Hausdorﬀ regular paratopological group. By Proposition 3.15, (G, τsr) is a pseudocompact periodic topological group. Let G˘ be the Ra˘ıkov completion of the topological group (G, τsr). The feeble compactness of (G, τsr) implies the precompactness of (G, τsr) and the compactness of G˘. We claim that the group G˘ is periodic. Suppose to the contrary that there exists a non- periodic element x ∈ G˘. Then for each positive integer n there exists a neighborhood Vn ∋ x n ˘ such that Vn 6∋ e. Being pseudocompact, the subgroup G of G intersects each nonempty ˘ Gδ-set in G. Consequently, there exists a point y ∈ G ∩ n∈ω Vn. By the periodicity of G, n T n there is a positive integer n such that y = e. But this contradicts e∈ / Vn . This contradiction n proves the periodicity of the group G˘. For each n> 1 we put G˘n = {x ∈ G˘ : x = e}. Since the compact topological group G˘ is Baire, there is a number n> 1 such that the set G˘n has the nonempty interior. Since the group G is dense in G˘, we see that there are a point x ∈ G and a neighborhood U of the identity of G such that (xu)n = e for each u ∈ U. Let y be an arbitrary point of G and V ∈ τ be an arbitrary neighborhood of the point y−1. Then xny−1 = y−1 ∈ V . By the continuity of the multiplication there is a neighborhood W ∈ τ of the identity such that W ⊆ U and (xW )n−1xy−1 ∈ V . Let z ∈ yW be an arbitrary point. Then (xy−1z)n = e and hence z−1 = (xy−1z)n−1xy−1 ∈ (xW )n−1xy−1 ⊆ V . Thus the inversion on the group (G, τ) is continuous and (G, τ) is a topological group. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 15

The following example shows that Proposition 3.17 cannot be generalized to T1 groups (even to sequentially pracompact and periodic). Moreover, in [1], Alas and Sanchis proved that each topologically periodic T0 paratopological group is T1. Example 3.18 combined with Proposition 3.17 show that there is a T1 periodic paratopological group G which is not Hausdorﬀ.

Example 3.18. There exists a T1 periodic sequentially pracompact paratopological group G which is not a topological group.

Proof. For each positive integer n let Cn be the cyclic group {0,...,n − 1} endowed with the discrete topology and the binary operation ⊕ such that x ⊕ y ≡ x + y mod n for any ∞ x,y ∈ Cn. Let G = L Cn be the direct sum of the cyclic groups Cn. Elements of G can be n=1 thought as functions x : N → ω such that x(n) < n for all n ∈ N and x(n) = 0 for all but ﬁnitely many n. Let F be the family of all non-decreasing unbounded functions from N to ω. For each f ∈ F put

Of = {0} ∪ {x ∈ G : ∃n ∈ N (0 nx(m) = 0)}.

It is easy to check that the family {Of : f ∈ F} is a base at the zero of a T1 semigroup topology τ on G. For each positive integer m ≥ 2 consider the function δm ∈ G such that −1 −1 δm (1) = {m} and δm (0) = N \ {m}. Let f, g ∈ F be arbitrary functions and let x ∈ G be an arbitrary element. There exists a number m such that f(m) ≥ 2, g(m) ≥ 2, and x(n) = 0 for each n ≥ m. Then δm ∈ Og and x + δm ∈ Of . Therefore x ∈ Of . Hence Of = G for each f ∈ F and the topology τ is not Hausdorff. Since any two nonempty open subsets of (G, τ) intersect, any two nonempty open subsets of each power of G in the box topology intersect, too. Therefore each power of G in the box topology is feebly compact. Put D = {0} ∪ {δm : m ≥ 2}. Then D witness that G is sequentially pracompact, because D is homeomorphic to a convergent sequence and dense in G. Problem 3.19. Suppose a Hausdorff topologically periodic paratopological group G satisfies one of the following conditions: • G is countably pracompact (and left ω-precompact), • G is feebly compact (and left ω-precompact). Is G a topological group? If a paratopological group G satisfies one of the conditions of Problem 3.19 and is quasiregular or saturated or Baire, then G is a topological group. We recall the following question by van Douwen [31] from 1980, which is, according to references from [48, p. 2], considered central in the theory of topological groups. Question 3.20. Is there an infinite countably compact group without non-trivial convergent sequences? For a history of related results see introductions of [97] and [48]. In particular, in 1976 Hajnal and Juhász in [45] constructed the first example of such a group under the Continuum Hypothesis. In 2004 Garc´ıa-Ferreira, Tomita, and Watson in [39] obtained such a group from a selective ultrafilter. In 2009 Szeptycki and Tomita in [ST] showed that in the Random model there exists such a group, giving the first example that does not depend on selective ultrafilters. Finally, Hruˇsák, van Mill, Ramos-Garc´ıa, and Shelah obtained a positive answer to van Douwen’s question in ZFC, see [48] for “a presentable draft of the proof”. The built 16 TARAS BANAKH AND ALEX RAVSKY group is Boolean and at p.17 the authors say that they do not know how to modify (in ZFC) their construction to get a non-torsion example and formulate the question about an existence of such a group in ZFC. We denote by TT a bit stronger axiomatic assumption: there is an infinite torsion-free Abelian countably compact topological group without non-trivial convergent sequences. The first example of such a group was constructed by Tkachenko [91] under the Continuum Hypothesis. Later, the Continuum Hypothesis was weakened to Martin’s Axiom for σ-centered posets by Tomita in [96], for countable posets in [51], and finally to the existence of c many incomparable selective ultrafilters in [62]. For a subset A of a group G by hAi⊆ G we denote the subgroup generated by the set A in G. The proof of Lemma 6.4 in [13] implies the following Lemma 3.21. Under TT, the free Abelian group G generated by the cardinal c admits a Hausdorff group topology τ such that for each countable infinite subset M ⊆ G there exists an τ ordinal α ∈ c ∩ M such that M ⊆ h[0, α)i. Example 3.22. Under TT there exists a functionally Hausdorff countably compact free Abelian paratopological group (G, σ), which is not a topological group. Proof. Let G be a free Abelian group generated by the cardinal c. Elements of the group G can be thought as functions x : c → Z with finite support supp(x) := {α ∈ c : x(α) 6= 0}. Each ordinal α ∈ c can be identified with its characteristic function δα : c → {0, 1} (such that −1 δα (1) = {α}). In the free Abelian group G, consider the subsemigroup S = {0}∪ x ∈ G \ {0} : x max(supp(x)) < 0 . By Lemma 3.21, the group G admits a Hausdorff group topology τ such that for each countable τ infinite subset M ⊆ G there exists an ordinal α ∈ c ∩ M such that M ⊆ h[0, α)i. It is easy to check that the family {U ∩ S : 0 ∈ U ∈ τ} is a base at the zero of a semigroup topology σ on G. Let M be an arbitrary countable infinite subset of the group G. By the choice of τ the topology τ, there exists an ordinal α ∈ M ∩ c such that M ⊆ h[0, α)i. We claim that α belongs to the closure of the set M in the topology σ. Given any open neighborhood U ∈ τ τ of zero, we need to prove that the set α + (U ∩ S) intersects M. Since α ∈ M , there exists an element µ ∈ M ∩ (α + U). On the other hand, µ ∈ M ⊆ h[0, α)i ⊆ α + S and hence µ ∈ (α + U) ∩ (α + S) = α + (U ∩ S). Therefore, (G, σ) is a countably compact Hausdorff paratopological group. In particular, the space (G, σ) is not discrete. Taking into account that S ∈ σ and S ∩ (−S)= {0} ∈/ σ, we see that (G, σ) is not a topological group.

Example 3.22 negatively answers Problem 1 from [43] under TT. Since each countably compact left ω-precompact paratopological group is a topological group [71, p. 82], we see that Example 3.22 implies the negative answers to Question A in [1] and to Problem 2 in [43] under TT. Problem 3.23. Is there a ZFC-example of a paratopological group G such that G is not a topological group but G satisfies one of the following conditions: (i) G is Hausdorff countably compact, (ii) G is Hausdorff countably pracompact Baire, (iii) G is T1 countably compact? If G is such a group, then G is not topologically periodic, not saturated, not quasiregular, and not left ω-precompact. Moreover, G×G is not countably compact. By Theorem 2.2 from ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 17

[1], the group (G, σ) in Example 3.22 is Baire. Therefore under TT there is a group G such that G satisfies all conditions listed in Problem 3.23. The following proposition was published in 2010 in the first arXiv version of this manuscript and answered Question B from [1]. In 2012 Sánchez generalized this result, obtaining a characterization of 2-pseudocompact T1 paratopological groups which are topological groups, see [80, Proposition 2.9]. This characterization was independently obtained by Xie and Lin in [59]. So we skip the proof of the proposition, but keep its statement for references farther in the paper. Also this and the next proposition positively answer a special case (for 2- pseudocompact groups) of Problem 8.1 in [5]. Proposition 3.24. Each 2-pseudocompact paratopological group of countable pseudocharacter is a topological group. In order to strengthen Proposition 3.24 for Hausdorff 2-pseudocompact paratopological groups we shall use the following lemmas. Lemma 3.25. [76, Theorem 0.5] A compact Hausdorff semigroup with separately continouous multiplication and two-side cancellations is a topological group. Lemma 3.26. Let K be a compact nonempty subset of a Hausdorff semitopological group G. Then the set GK = {x ∈ G : xK ⊆ K} is a compact Hausdorff topological group. −1 Proof. Observe that GK = Tx∈K Kx is a compact cancellative subsemigroup of the Haus- dorff semitopological group G. By Lemma 3.25, GK is a topological group.

Remark 3.27. Lemma 3.26 does not necessarily hold for T1 paratopological groups, as the following example shows. Let G = (Z, +) be the group of integers endowed with a topology with the base {{x} ∪ {z ∈ Z : z ≥ y} : x,y ∈ Z}. It is easy to check that G is a T1 paratopological group. Put K = {x ∈ Z : x ≥ 0}. Then K is a compact subset of the group G and GK = K is not a group. Lemma 3.28. [69, Proposition 1.8] If G is a paratopological group, K is a compact subset of G, F is a closed subset of G and K ∩ F = ∅, then there exists a neighborhood V ⊆ G of the identity such that V K ∩ F = ∅. Proposition 3.29. Each 2-pseudocompact Hausdorﬀ paratopological group containing a nonempty compact Gδ-set is a topological group.

Proof. Let (G, τ) be such a group and K be a nonempty compact Gδ-set of the space (G, τ). Suppose (G, τ) is not a topological group; then there exists a neighborhood U ∈ τe such that −1 −1 2 V 6⊆ U ⊆ (U ) for each neighborhood V ∈ τe. By induction using Lemma 3.28 we can 2 build a sequence {Vi}i∈ω ⊆ τ of neighborhoods of e such that V0 = U, Vi+1 ⊆ Vi for each i ∈ ω and Ti∈ω ViK = K. Put H = Ti∈ω Vi. The choice of the sequence (Vi)i∈ω ensures −1 −1 −1 −1 that H is a semigroup. Moreover, since Vi+1 ⊆ Vi+1Vi+1 ⊆ Vi for each i ∈ ω, we see that −1 H is a closed subset of G. Moreover, HK ⊆ Ti∈ω ViK = K, so H ⊂ GK. Since GK is −1 −1 a group (see Lemma 3.26), H ⊆ GK too, so H is a compact cancellative topological −1 −1 semigroup. By Lemma 3.25, H is a group. We have Vi 6⊆ U for each i ∈ ω. Since the −1 −1 group G is 2-pseudocompact, there is a point x ∈ G such that xW ∩ (Vi\U ) 6= ∅ for each −1 −1 −1 −1 W ∈ τe and each i > 0. Then xW ∩ (Vi\U ) 6= ∅ and Wx ∩ (Vi \U) 6= ∅. Therefore −1 −1 −1 −1 −1 −1 −1 −1 x ∈ Vi ⊆ Vi−1. Hence x ∈ H . But H = H ⊆ U ∩ U and U ∩ (V1\U ) = ∅. This contradiction proves that G is a topological group. 18 TARAS BANAKH AND ALEX RAVSKY

The authors do not know whether counterparts of Proposition 3.29 hold for T1 2-pseudocompact or countably compact paratopological groups. But a T0 sequentially compact paratopological group GS from Example 5.27 contains an open compact subsemigroup S. Nevertheless, GS is not a topological group. The following Example 3.30 (which is an extension of [71, Example 5.17]) shows that counterparts of Proposition 3.24 and Implication (4) ⇒ (1) in Proposition 3.15 do not hold for sequentially pracompact paratopological groups. Also Example 3.30 negatively answers Problem 6.2 in [92] and Problem 8.1 in [5]. Example 3.30. There exists a Hausdorff second countable sequentially pracompact paratopological group G which is not a topological group. Proof. Let T = {z ∈ C : |z| = 1} be the unit circle endowed with the standard Euclidean topology τ and the group operation of multiplication of complex numbers. Fix a surjective homomorphism s : T → Q. For every real number a consider the subsets > ≥ Ta := {1} ∪ {x ∈ T : s(x) > a} and Ta := {1} ∪ {x ∈ T : s(x) ≥ a} and the familes of sets ′ > ′ ≥ ′ > ′ > B1 = {Ta : a< 0}, B2 = {T0 }, B3 = {T0 } and B4 = {Ta : a> 0}. ′ It is easy to check that for every i ∈ {1, 2, 3, 4} the family Bi is a base at the identity of a first-countable semigroup topology on the group T. Consequently, for every i the family Bi = ′ {S∩U : S ∈Bi, 1 ∈ U ∈ τ} is a base at the identity of a first-countable functionally Hausdorff semigroup topology on the group T. Denote this topology by τi and put Gi := (T,τi). Clearly, τ ⊂ τ1 ⊂ τ2 ⊂ τ3 ⊂ τ4. Fix any element d with s(d) > 0 and put D = {nd : n ∈ N}. It is easy to check that the set D is dense in G4. Let {xn}n∈ω be any sequence of points of D. Since the group (G, τ) is sequentially compact, there exists a point x ∈ T and a subsequence {xn(k)}k∈ω converging to x in (T,τ). Taking a subsequence, if necessary, we can assume that s(xn(k+1)) ≥ s(xn(k))+ s(d) for each k. For any a > 0 there exists K > 0 such that > s(xn(k)) >s(x)+ a and so xn(k) ∈ x + Ta for each k > K. It follows that {xn(k)} converges to x in G4. It is easy to check that the group (T,τ) is pseudobounded, the group G1 is ω- pseudobounded, and the groups G2, G3, and G4 are not ω-pseudobounded. Observe that for −1 −1 −1 i ∈ {1, 2} the preimege s (0) is a subgroup of countable index in T and τi↾s (0) = τ↾s (0). Since the space (T,τ) is second countable, the paratopological group Gi has a countable network. Since the space Gi is first-countable, by [69, Proposition 2.3], the paratopological group Gi is second countable. We claim that for i ∈ {2, 3} the group Gi does not have a countable network consisting of closed subsets. To derive a contradiction, assume that N is a countable family of closed subsets of Gi such that for any x ∈ Gi and neighborhood Ox ⊆ Gi of x there exists a set N ∈ N with x ∈ N ⊆ Ox. In particular, for any x ∈ T there exists a ≥ closed set Nx ∈ N such that x ∈ Nx ⊆ x + T0 . We claim that the set Nx is nowhere dense in T. To derive a contradiction, assume that the τ-closure of Nx has nonempty interior in T. τ Since the set {nd : n τi ≥ s(z) ≥ s(x) > s(y). Then z ∈ Nx ∩ V ∩ (y + T0 ) and hence y ∈ Nx = Nx ⊆ x + T0 and s(y) ≥ s(x), which contradicts the choice of y. This contradiction shows that the set Nx is nowhere dense in T. Then the union T = Sx∈X Nx is a meager subset in T, which contradicts the Baireness of the compact space T. This contradiction proves that the space Gi has no > countable network consisting of closed subsets. Observe that the set T\T0 is closed and ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 19

discrete in G4. Thus the space G4 has extent e(G4)= c. The set {(x, −x) : x ∈ T} is a closed discrete subset of G3 × G3, which implies that the space G3 × G3 has extent e(G3 × G3)= c. The values of the extent e(G3) and the Lindel¨of number l(G3) of the space G3 depend on the homomorphism s. Claim 3.31. There exists a surjective homomorphism s : T → Q such that the set s−1(0) contains a compact subset K of T of cardinality |K| = c. In this case e(G3)= l(G3)= c. Proof. By [50, 19.2], the real line contains a Cantor set C ⊆ R, which is linearly independent 2π over the ﬁeld Q. Replacing C by c C where c ∈ C is any point, we can assume that 2π ∈ C. Replacing C by a suitable closed subset, we can additionally assume that the Q-linear hull of C does not contain some real number r. Then the set C ∪ {r} is linearly independent over Q and by Zorn’s Lemma can be enlarged to a maximal linearly independent set B. Let f : R → Q be a unique Q-linear functional such that f(r)=1and f(B \{r})= {0}. Consider the homomorphism h : R → T, h : x 7→ eix, and observe that h−1(1) = 2πZ ⊆ f −1(0). Then there exists a unique homomorphism s : T → Q such that f = s ◦ h. The surjectivity of f implies the surjectivity of s. Since h has countable preimages of points, the image K = h(C) is an uncountable compact subset of T with s(K) = s(h(C)) = f(C) = {0}. By [50, 3.3.2 c > and 6.5], |K| = . Since τ ⊆ τ3, the set K is a closed in G3. Since (x + T0 ) ∩ K = {x} for every x ∈ K, the set K is discrete in G3. Consequently c ≤ e(G3) ≤ l(G3)= c. Claim 3.32. There is a surjective homomorphism s : T → Q such that inf s(F ) = −∞ for every uncountable closed subset F ⊆ T. In this case e(G3)= l(G3)= ω. Proof. The construction of s resembles the construction of a Bernstein set. Since the family F of uncountable closed subsets of the real line has cardinality continuum, it can be enumerated as F = {Fα}α |ω + α| = |L<α|. After completing the inductive construction, we obtain a linearly independent subset {2π} ∪ {xα}α

Now we prove that the space G3 is Lindel¨of. Given any open cover U of the space G3, for every > point x ∈ G3 choose a neighborhood Ox ∈ τ of x and a set Ux ∈U such that (x+T0 )∩Ox ⊆ Ux. > For any rational number a, consider the subsets Aa := {Ox : x ∈ T\Ta } and Ba := T\Aa ⊆ > > S > Ta . Since Ba ⊆ Ta is a closed subset of T with inf s(Ba) ≥ inf s(Ta )= a, the choice of the homomorphism s ensures that the set Ba is at most countable. Since the metrizable compact > space (T,τ) is hereditarily Lindel¨of, there exists a countable subset Ca of T\Ta such that Aa = S{Ox : x ∈ Ca}. Consider the countable subfamily V := Sa∈Q{Ux : x ∈ Ba ∪ Ca} of U. We claim that T = S V. Indeed, given any point y ∈ T, choose a negative rational number a

> and hence y ∈ Ox for some x ∈ Ca ⊆ T \ Ta . It follows that s(x) ≤ a y ∈ Ox ∩ (x + T0 ) ⊆ Ux ∈ V. Therefore the space G3 is Lindel¨of and has Lindel¨of number l(G3) ≤ ω. Since ω ≤ e(G3) ≤ l(G3) ≤ ω we have e(G3)= l(G3)= ω.

In [81, Theorem 1] Tkachenko and Sanchis constructed a Hausdorff 2-pseudocompact Fréchet-Urysohn paratopological group which is not a topological group, thus answering a second author’s question from an older version of this manuscript. The next proposition was motivated by a question of Gutik whether the closure of a subgroup of a countably compact paratopological group is a group. Also it provides a partial solution to Problem 4.4 from [95]: Let G be a Hausdorff feebly compact paratopological group such that the closure of every subgroup of G is a subgroup. Is G a topological group?

Proposition 3.33. Let G be a countably compact paratopological group such that the closure of each cyclic subgroup of G is a group. Then G is a topological group.

Proof. Let x be an arbitrary point of G and H be the closure of the cyclic subgroup hxi generated by x. By our assumption, H is a subgroup of G. We claim that the group H is left ω-precompact. It suﬃces to show that H ⊆ hxi · U for any neighborhood U ⊆ H of the identity x0. For every point y ∈ H the density of the subgroup hxi in H yields a point z ∈ hxi∩ Uy−1. Then y ∈ z−1U ⊆ hxi · U. The space H is countably compact being a closed subspace of a countable compact space G. Therefore the group H is 2-pseudocompact. By Lemma 3.7 H is Baire and by Proposition 3.13 H is feebly compact. Thus by Proposition 3.15 H is topologically periodic, and so is the paratopological group G. By [17], G is a topological group.

The following proposition answers a question of Guran.

Proposition 3.34. Each Hausdorﬀ σ-compact feebly compact paratopological group is a compact topological group.

Proof. Let (G, τ) be such a paratopological group and (G, τsr) be its semiregularization. By Lemma 1.1, (G, τsr) is a feebly compact Hausdorﬀ regular paratopological group and by Proposition 3.15, (G, τsr) is a pseudocompact topological group. Write the σ-compact space G as the countable union G = Sn∈ω Kn of compact subsets Kn, which remain compact in the topological group (G, τsr). Since the pseudocompact topological group (G, τsr) is 2- pseudocompact, it is Baire by Lemma 3.7, so for some n ∈ ω the compact set Kn has nonempty interior in (G, τsr) and also in (G, τ). This means that the space (G, τ) is locally compact. By Ellis’ Theorem (see Subsection 1.6) (G, τ) is a topological group. Being feebly compact, the locally compact (and hence Ra˘ıkov complete) topological group (G, τ) is compact.

4. Preservation of feebly compactness by some operations over paratopological groups In this section we prove that the feeble compactness of paratopological groups is preserved by Tychonoﬀ products and extensions. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 21

4.1. Tychonoff products of feebly compact paratopological groups. By Tychonoff’s theorem, the Tychonoff product of any family of compact spaces is compact. The productivity of other compact-like properties can be a non-trivial problem. For instance, the product of a countable family of sequentially compact spaces is sequentially compact [36, Theorem 3.10.35]. But the Cantor cube Dc is not sequentially compact (see [36], the paragraph after Example 3.10.38). On the other hand some compact-like spaces are also preserved by products, see [98, § 3–4, 7] (especially Theorem 3.3, Proposition 3,4, Example 3.15, Theorem 4.7, and Example 4.15), [84, § 5], and [44, 2.3]. In [89], [90], and [93] Tkachenko considered the productivity of various properties of topological groups. All but one non-productive properties of spaces listed in [93, p. 2], are (possible, in models) also non-productive for topological groups. For instance, Novák [66] and Teresaka [86] constructed examples of two countably compact spaces whose product is not pseudocompact, see also [36, 3.10.19]. Using Martin’s Axiom, van Douwen [31] constructed countably compact topological groups G, H whose product G×H is not countably compact. Later, under MAcountable Hart and van Mill [46] constructed a countably compact topological group G whose square G×G is not countably compact. As far as the authors know, there is no ZFC-example of a family {Gα : α ∈ A} of countably compact topological groups whose Tychonoff product Q {Gα : α ∈ A} is not countably compact, see the surveys [24] and [25]. Feeble compactness of products of spaces is preserved in particular cases, see [82]. Also Dow et al. in Theorem 4.1 of [32] proved that the Tychonoff product of a family of sequentially feebly compact spaces is again sequentially feebly compact, and in Theorem 4.3 that every product of feebly compact spaces, all but one of which are sequentially feebly compact, is feebly compact. Comfort and Ross in [27, Theorem 1.4] proved that the product of any family of pseudocompact topological groups is pseudocompact. The main result of this subsection is a generalization of this result to paratopological groups. Theorem 4.1. The Tychonoff product of an arbitrary family of feebly compact paratopological groups is feebly compact.

Proof. Let (G, τ) be the Tychonoff product of a family {(Gα,τα) : α ∈ A} of feebly compact paratopological groups. It is easy to see that the semiregularization (G, τsr) of the space (G, τ) coincides with the Tychonoff product Qα∈A(G, ταsr) of the semiregularizations of the spaces (Gα,τα). Let τsre be the family of all open neighborhoods of the identity in the paratopological group (G, τ ). It is easy to see that the intersection H := U ∩ U −1 sr TU∈τsre is a normal subgroup of G. Let π : G → G/H be the quotient map. For any element α ∈ A let ταsre be the family of all open neighborhoods of the identity eα of the paratopological group (G ,τ ), H = U ∩ U −1, π : (G ,τ ) → (G ,τ )/H be the quotient α αsr α TU∈ταsre α α αsr α αsr α map, and pα : (G, τsr) → (Gα,ταsr) be the coordinate projection. Let π : Qα∈A(Gα,ταsr) → Qα∈A(Gα,ταsr)/Hα be the product of the family of quotient maps {πα}α∈A. By Proposition τsr τsr −1 ταsr 2.3.29 from [36], the map π is open. Since H = {e} ∩ ({e} ) and Hα = {eα} ∩ ταsr −1 −1 ({e} ) for each α ∈ A, we see that H = Qα∈A Hα = Tα∈A πα (Hα) = ker π. Let i : (G, τsr)/H → Qα∈A(Gα,ταsr)/Hα be a map such that i(xH) = π(x) for each x ∈ G. Theorem on continuous epimorphisms [69, p. 42] implies that the map i is well-defined and is a topological isomorphism. Let α ∈ A be any element. Then (Gα,ταsr) is a regular feebly compact paratopological group (see the remark after Lemma 3.3). So (Gα,ταsr) is a feebly compact topological group by Lemma 3.11. Then the group (Gα,ταsr)/Hα is a pseudocompact topological group. Therefore by Theorem 1.4 from [27], the product Qα∈A(Gα,ταsr)/Hα is pseudocompact and so is its isomorphic copy (G, τsr)/H. Consider the quotient map −1 q : (G, τsr) → (G, τsr)/H. By Lemma 3.1, each open subset U ∈ τsr is equal to q (q(U)). 22 TARAS BANAKH AND ALEX RAVSKY

This fact and the feeble compactness of the quotient group (G, τsr)/H implies the feeble compactness of the topological group (G, τsr). Finally, by Lemma 3.5, the paratopological group (G, τ) is feebly compact. Problem 4.2. Is the countably pracompactness or 2-pseudocompactness of paratopological groups preserved by Tychonoff products? This problem is open since 2010, from the first version of this manuscript. Recently it obtained partial negative answers. Namely, Garc´ıa-Ferreira and Tomita in [40] under CH constructed a countably compact topological group whose square is not selectively pseudocompact and therefore not countably pracompact. Bardyla, Ravsky, and Zdomskyy [16] constructed under MA a Boolean Tychonoff countably compact topological group without non-trivial convergent sequences whose square is not countably pracompact. 4.2. Extensions of feebly compact paratopological groups. A (topological, algebraic, or topological-algebraic) property P of a semitopological group is called a three-space property provided whenever N is a closed normal subgroup of a group G and both N and G/N have P, the group G also has P. Various three-space properties in the class of topological groups are investigated for almost a century. Their list is quite long and includes compactness [38], local compactness [83], Ra˘ıkov completeness, precompactness, pseudocompactness [26], connected- ness, and metrizability [41], see also [26], [28], and [47]. On the other hand, in [22] Bruguera and Tkachenko constructed an Abelian topological group G and a closed subgroup N of G such that the groups N and G/N are countably compact, but G is not. Also they studied a lot of three-space properties for compact-like sets in [21]. Lin, Lin, and Xie in [58] considered some specific three-space properties. Recently three-space properties were investigated in some classes of paratopological and semitopological groups, and not all of three-space properties of topological groups remain three-space also for these wider classes of groups. Among such properties let us mention the metrizability (even for Tychonoff paratopological groups, [52, Example 3.3]), the first- and second-countability [37]. On the other hand, to be a topological group is a three-space property in the class of paratopological groups [73, Lemma 4]. Since by [71, Proposition 5.5], a locally compact paratopological group is a topological group, it follows that compactness and local compactness are three-space properties in the class of paratopological groups. Similarly to the proof of Theorem 3 in [55], we can show that both the pseudoboundedness and the ω-pseudoboundedness are three-space properties in the class of topologized groups. In [71, p.50-51] the second author showed that for each infinite cardinal λ both ψ(G) ≤ λ and d(G) ≤ λ are three-space properties in the class of paratopological groups. For more three-space properties in the class of paratopological groups see [60]. The metrizability, first-countability and second-countability also are not three-space properties in the class of quasitopological groups, see [57]. Some three-space properties of semitopological groups are also considered in [42] and [53]. The main result of this subsection is Theorem 4.8 establishing the three-space property of the feeble compactness in the class of paratopological groups. It generalizes the following result (see Theorem 6.3(c) from the paper [26] by Comfort and Robertson). Theorem 4.3. Let H be a closed normal subgroup of a Hausdorff topological group G. If the topological groups H and G/H are feebly compact, then so is the topological group G. To generalize Theorem 4.3 to the class of paratopological groups, we need to make some preliminary job. First we observe that the closedness of the subgroup H can be easily removed from Theorem 4.3. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 23

Lemma 4.4. Let H be a normal subgroup of a Hausdorﬀ topological group G. If the topological groups H and G/H are feebly compact, then so is the topological group G. Proof. Observe that the closure H¯ of the group H in G is feebly compact (since it contains a dense feebly compact subspace). By the continuity of the homomorphism h : G/H → G/H¯ , h : gH 7→ gH¯ , the quotient group G/H¯ is feebly compact, being a continuous image of the feebly compact space G/H. Applying Theorem 4.3, we conclude that the topological group G is feebly compact. Lemma 4.5. Let G be a regular paratopological group and H be a compact normal subgroup in G. Then (1) the quotient paratopological group G/H is regular; (2) if G/H is a topological group, then G is a topological group. Proof. 1. To prove the regularity of G/H, it suﬃces to show that every open neighborhood U ⊆ G/H of the identity contains the closed neighborhood of the identity. Let π : G → G/H be the quotient homomorphism. Then π−1(U) = π−1(U)H is an open neighborhood of the identity. By the regularity of G, it contains a closed neighborhood V of the identity. Then VH ⊆ π−1(U)H. Propositions 1.12 and 1.13 from [69] imply that the map π is open and closed. Then π(VH) = π(V ) ⊆ U is a closed neighborhood of the identity in G/H, which follows that the paratopological group G/H is regular. 2. By Lemma 5.4 from [71] or its counterpart in English in [74], or by Proposition 3.2, H is a topological group. Now by Lemma 4 from [73], G is a topological group.

Corollary 4.6. If G is a regular paratopological group then its T0-reflexion T0G is regular too. Lemma 4.7. Let (G, τ) be a regular Hausdorff paratopological group and H be a normal subgroup of G. If the spaces H and G/H are feebly compact, then so is the space G. Proof. Let G˘ = (G,˘ τ˘) be the Ra˘ıkov completion of the paratopological group (G, τ) and G˘sr = (G,˘ τ˘sr) be the semiregularization of the paratopological group G˘. By Theorem 2.6, ♯ τ =τ ˘sr↾G and the group coreflexion of G˘sr coincides with the Ra˘ıkov completion G˘ of the group coreflexion G♯ of the paratopological group G. By Proposition 3.15, the regular feebly compact paratopological group (H,τ↾H) is a topological group. Then τ ♯↾H = τ↾H and the topological group (H,τ ♯↾H) is feebly compact and precompact by Lemma 3.8. Then its closure H¯ in the Ra˘ıkov-complete topological group G˘♯ is compact, see [7, 3.7.9]. Since the ♯ identity map G˘ → G˘sr is continuous and the paratopological group G˘sr is Hausdorff (by Theorem 2.6), the subgroup H¯ is compact and closed in G˘sr. The normality of the subgroup H in G implies the normality of the group H¯ in the topological group G˘♯. Then the set Γ = GH¯ is a subgroup of G˘sr. Consider the quotient paratopological group Γ/H¯ (which is regular by Lemma 4.5), and the surjective homomorphism π : G/H → Γ/H¯ assigning to each coset gH ∈ G/H the coset gH¯ ∈ Γ/H¯ . The continuity of the identity embedding G → GH¯ = Γ ⊆ G˘sr implies the continuity of the map π : G/H → Γ/H¯ . Then the space Γ/H¯ is feebly compact, being a continuous image of the feebly compact space G/H. By Proposition 3.15, the regular feebly compact paratopological group Γ/H¯ is a topological group, and by Lemma 4.5(2), the paratopological group Γ is a topological group and so is its subgroup G (here we use the fact that τ =τ ˘sr↾G established in Theorem 2.6). By Lemma 4.4, the topological group G is feebly compact. 24 TARAS BANAKH AND ALEX RAVSKY

Theorem 4.8. Let H be a normal subgroup of a paratopological group G. If the paratopological groups H and G/H are feebly compact, then G is feebly compact, too.

Proof. Consider the semiregularization Gsr of the paratopological group G and its T0-reflextion Γ := T0Gsr, which is a Hausdorff regular paratopological group (according to Lemma 1.1). Let π : G → Γ be the composition of the identity map G → Gsr and the quotient map Gsr → Γ. The normality of the subgroup H implies the normality of the subgroup N := π(H) in the group Γ. Being a continuous image of the feebly compact space H, the space N is feebly compact. Consider the homomorphism h : G/H → Γ/N assigning to each coset gH the coset π(gH) = π(g)N. The continuity of the map π : G → Γ implies the continuity of the homomorphism h. Then the feeble compactness of the quotient group G/H implies the feeble compactness of the quotient group Γ/N. By Lemma 4.7, the paratopological group Γ= T0Gsr is feebly compact. Now Lemma 3.1 implies that the space Gsr is feebly compact and Lemma 3.5 implies that so is the space G. Remark 4.9. Theorem 4.8 solves positively Problem 5.3 and the quasiregular case of Prob- lem 5.4 of Tkachenko [92]. As far as the authors know, Problem 5.7 of [92] whether 2- pseudocompactness is a three-space property in the class of (Hausdorff, T3) paratopological groups remains open.

5. Cone topologies In this section we systematically study the cone topologies on topological groups. Such topologies yield many pathological examples of paratopological groups, see e.g. [9], [14], [69], [70] or [87]. The cone topologies τS and τS~ on a paratopological group (G, τ) are determined by a normal subsemigroup S ⊆ G containing the identity e. The normality of S means that −1 g Sg = S for any element g ∈ G. Let ZS = Tx∈S{z ∈ S : xz = zx} be the center of the semigroup S. By definition, the cone topology τS on G consists of the sets U ⊆ G such that for every x ∈ U there exists an open neighborhood V ∈ τ of x such that V ∩ xS ⊆ U. The shifted cone topology τS~ on G consists of the sets U ⊆ G such that for every x ∈ U there exists an open neighborhood V ∈ τ of x and a point z ∈ ZS such that V ∩ xzS ⊆ U. It can be shown that (G, τS ) and (G, τS~ ) are paratopological groups. A particular case of this construction appears when G is a real linear space and S is a cone in G, that is a subset of G such that λx + µy ∈ S for any points x,y ∈ S and non-negative real numbers λ, µ, which explains the term “cone topology”. The Sorgenfrey topology on the real line R is a cone topology τS determined by the cone S = [0, +∞). If the topology τ on G is antidiscrete, then the paratopological groups (G, τS) and (G, τS~ ) will be denoted by GS and GS~ , respectively. Observe that for any semigroup topology τ on G, the identity maps (G, τS ) → GS and (G, τS~ ) → GS~ are continuous. So if a topological property P is preserved by continuous isomorphisms (for instance, the compactness, countably compactness, feeble compactness) and we wish the group (G, τS ) or (G, τS~ ) to have the property P, then we must assure the group GS or GS~ to have the property P, too. So we start with studying the paratopological groups GS and GS~ . As expected, in this basic case the cone topologies have very weak separation properties, and even T1 or T2 cases are degenerated, see Subsections 5.1 and 5.2. But non-Hausdorffness of these constructions will be compensated by a Hausdorff topology τ in Subsection 5.3. It turned out that there is a simple interplay between the algebraic properties of the semigroup S and compact-like properties of the paratopological groups GS and GS~ , see Subsections 5.1 and 5.2. In these subsections we assume that G ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 25

is an Abelian group and S ∋ 0 is a subsemigroup of G. We restrict our attention here to Abelian groups, because authors’ experience suggests that when we check an implication of topological properties for a paratopological groups, then positive results are proved for all groups, but negative are illustrated by Abelian counterexamples. In the sequel we shall need an elementary lemma. Lemma 5.1. If there is an element a ∈ S such that S ⊆ a − S, then S is a group. Proof. Since 2a ∈ S ⊆ a − S, there exists an element a′ ∈ S such that 2a = a − a′. Thus −a = a′ ∈ S. Since S is a subsemigroup of a group, it suﬃces to remark that for each x ∈ S we have −x ∈−S ⊆ S − a = S + a′ ⊆ S.

5.1. The paratopological groups GS. In this section we consider the interplay between algebraic properties of the subsemigroup S ∋ 0 of an Abelian group G and topological properties of the paratopological group GS whose topology consists of the sets A+S where A ⊆ G is an arbitrary subset. It is easy to see that the closure S of the semigroup S is equal to the clopen subgroup S − S of GS. The following propositions and lemmas can be easily derived from the deﬁnition of the topology of the paratopological group GS .

Proposition 5.2. The group GS is T0 iﬀ S ∩ (−S)= {0}.

Proposition 5.3. The group GS is T1 iﬀ S = {0}.

Lemma 5.4. A subset K of GS is compact iﬀ K ⊆ F + S for some ﬁnite subset F ⊆ K.

Lemma 5.5. For each subset A ⊆ G its closure in GS coincides with the set A − S. It is easy to prove the following characterization.

Proposition 5.6. The paratopological group GS is topologically periodic iff S is a subgroup of the group G and the quotient group G/S is periodic. Next, we characterize feebly compact and countably pracompact spaces among paratopological groups GS. Proposition 5.7. The following conditions are equivalent: (1) The space GS is feebly compact. (2) The space GS is countably pracompact. (3) The space GS is sequentially pracompact. (4) The subgroup S − S has finite index in G. Proof. The implications (3) ⇒ (2) ⇒ (1) are trivial. (4) ⇒ (3) Since the subgroup S − S has a finite index in G then there exists a finite set F ⊆ G such that F + S − S = G. Thus F + S is dense in GS. Any sequence of points of F + S has a subsequence contained in a set f + S for some f ∈ F , so it converges to f. (1) ⇒ (4) If the subgroup H = S − S has infinite index in G, then we can find an infinite subset I ⊆ G such that the indexed family {xH}x∈I is disjoint. Since {xH}x∈I is an infinite locally finite family of open sets in GS, the space GS is not feebly compact. Proposition 5.8. The following conditions are equivalent: (1) The space GS is compact; (2) The space GS is ω-bounded. (3) The space GS is totally countably compact. 26 TARAS BANAKH AND ALEX RAVSKY

(4) The paratopological group GS is precompact. (5) The semigroup S is a subgroup of G of ﬁnite index.

Proof. The implications (5) ⇒ (1) ⇒ (2) ⇒ (3) are trivial. (3) ⇒ (4) If the space GS is totally countably compact then the set {0} = −S is compact and hence is contained in F + S for some finite subset F of −S. Since GS is feebly compact, by Proposition 5.7, the subgroup S − S has finite index in G and hence G = D + S − S for some finite set D ⊆ G. Thus G ⊆ D + S − S ⊆ D + S + F + S ⊆ D + F + S, which follows that the paratopological group GS is precompact. (4) ⇒ (5) Assuming that the paratopological group GS is precompact, choose a finite subset F of G such that F + S = G. Then F + (S − S) = G and the subgroup H = S − S has finite index in G. For each element x of the finite set F ∩ H ⊆ H = S − S find elements yx, zz ∈ S such that x = yx − zx. Consider the sum z := Px∈F ∩H zx ∈ S and observe that yx − zx ∈ S − z + S = −z + S and hence S − S = (F ∩ H)+ S ⊆−z + S + S = −z + S, which implies that S − S = z + S − S ⊆ S and S = S − S is a subgroup (of finite index in G). Proposition 5.9. The following conditions are equivalent:

(1) The space GS is sequentially compact. (2) The space GS is countably compact. κ (3) A power GS of the space GS is countably compact for each cardinal κ. (4) The space GS is countably o-compact. (5) The paratopological group GS is 2-pseudocompact. (6) The subgroup S − S has finite index in G and each countable subset C ⊆ S − S is contained in a − S for some a ∈ S; (7) The subgroup S −S has finite index in G and each countable subset C ⊆ S is contained in a − S for some a ∈ S; (8) The subgroup S − S has finite index in G and each infinite subset C ⊆ S has infinite intersection with the set a − S for some a ∈ S; (9) Each infinite subset C ⊆ G has infinite intersection with the set a−S for some a ∈ G. Proof. We shall prove the implications (1) ⇒ (2) ⇒ (5) ⇒ (9) ⇒ (8) ⇒ (7) ⇒ (6) ⇒ (1), (9) ⇒ (4) ⇒ (5), and (6) ⇒ (3) ⇒ (2). Among those implications, (1) ⇒ (2) ⇒ (5), (4) ⇒ (5) and (3) ⇒ (2) are trivial. (5) ⇒ (9) Let C be a countable infinite subset of G

and C = {xn}n∈ω be an enumeration of C such that xn 6= xm for any numbers n

Let C = {cn}n∈ω ⊆ S be a countable subset of S. Consider the set D = {dn : n ∈ ω}, where n dn = Pi=0 ci for each n ∈ ω. If the set D if ﬁnite, then C ⊆ (Pd∈D d) − S. Now suppose ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 27

the set D is infinite. By (8), there are an element a ∈ S and an infinite subset B of D such that B ⊆ a − S. Therefore for each number i ∈ ω there exists a number n ∈ ω such that i ≤ n and dn ∈ B. Then ci ∈ dn − S ∈ a − S − S ∈ a − S. (7) ⇒ (6). Let {cn : n ∈ ω} be a countable subset of the subgroup S − S. Fix sequences {an : n ∈ ω} and {bn : n ∈ ω} of S such that cn = bn − an for each n. By (7), there exist an element a ∈ S such that bn ∈ a − S for each n. Then cn ∈ a − S for each n. (6) ⇒ (1) To show that the space GS is sequentially compact, fix any sequence {xn}n∈ω in G. By (6), the subgroup S − S has finite index in G and hence G = (S − S)+ F for some finite subset F . By the Pigeonhole Principle, for some z ∈ F the set Ω := {n ∈ ω : −xn ∈ S − S + z} is infinite. By the condition (6), the set {−xn − z}n∈Ω ⊆ S − S is contained in the set a − S for some a ∈ S. Then for every n ∈ Ω we have −xn − z ∈ a − S and hence xn ∈ −a − z + S, which means that −a − z is the limit of the (convergent) subsequence (xn)n∈Ω of the sequence (xn)n∈ω, and the space GS is sequentially compact. (9) ⇒ (4) To show that the space GS is countably o-compact, fix a non-increasing sequence (Un)n∈ω of nonempty open sets in GS . For every n ∈ ω fix a point un ∈ Un and observe that un + S ⊆ Un. By the condition (9), there exists a point a ∈ G such that the set Ω = {n ∈ ω : un ∈ a − S} is infinite. Then a ∈ un + S ⊆ Un for every n ∈ Ω and hence a ∈ Tn∈Ω Un = Tn∈ω Un. (6) ⇒ (3) Let κ be an arbitrary cardinal. By (6) the subgroup H = S − S has finite index in G, which implies that the space GS is homeomorphic to the product HS × D of the paratopological group HS with a finite discrete space D. Then κ κ κ κ the power GG is homeomorphic to HS × D . By (6), for any sequence {xn}n∈ω in H there κ κ exists a point a ∈ H such that {xn}n∈ω ⊆ a + S , which implies that a is a limit of the κ κ κ sequence (xn)n∈ω. Therefore, the space HS is sequentially compact and the product HS × D κ is countably compact and so is its topological copy GS. Proposition 5.10. The following statements are equivalent:

(1) Every sequence in GS is convergent. κ (2) A power GS of the space GS is sequentially compact for each cardinal κ. c (3) The power GS is sequentially compact. (4) Each countable subset C ⊆ G is contained in a − S for some a ∈ S. (5) G = S − S and each countable subset C ⊆ S is contained in a − S for some a ∈ S. (6) G = S − S and each infinite subset C ⊆ S has infinite intersection with the set a − S for some a ∈ S. (7) G = S − S and each infinite subset C ⊆ G has infinite intersection with the set a − S for some a ∈ G. Proof. The implication (4) ⇒ (1) ⇒ (2) ⇒ (3) are trivial and the equivalences (4) ⇔ (5) ⇔ (6) ⇔ (7) follow from the corresponding equivalences in Proposition 5.9. It remains to prove c that (3) ⇒ (7). Assuming that the power of GS is sequentially compact, we first prove that the subgroup H = S −S coincides with the group G. Assuming that G 6= H, we conclude that the space GS is homeomorphic to the product HS × D for some discrete space D containing more than one point. The power Dc contains a closed topological copy of the Stone-Cechˇ c compactification βω and hence is not sequentially compact. Since the space GS containst c c a closed topological copy of the space D , the space GS is not sequentially compact, which c contradicts (3). This contradiction shows that G = S − S. The sequential compactness of GS implies the sequential compactness of the space GS. Now Proposition 5.9 ensures that the condition (7) holds. 28 TARAS BANAKH AND ALEX RAVSKY

Finally, let us observe that the following trivial characterization of the ﬁnal compactness of the spaces GS.

Proposition 5.11. The space GS is Lindel¨of iﬀ the paratopological group GS is left ω- precompact.

5.2. The paratopological groups GS~ . In this section we survey some compact-like properties of the paratopological groups GS~ . Here G is an Abelian group, S ∋ 0 is a subsemigroup of G and GS~ is the group G endowed with the topology τS~ consisting of all sets U ⊆ G such that for every u ∈ U there exists x ∈ S such that u + x + S ⊆ U. The topology τS~ will be called the a shift cone topology generated by the subsemigroup S of G. It is easy to check that the closure S of the semigroup S is equal to the subgroup S − S, which is closed-and-open in GG~ . Let us also observe that τS ⊆ τS~ .

Proposition 5.12. The space GS~ is Hausdorﬀ iﬀ S = {0}.

Since τS ⊆ τS~ , Proposition 5.6 implies the following characterization.

Proposition 5.13. The group GS~ is topologically periodic iﬀ S is a subgroup of the group G and the quotient group G/S is periodic. Proposition 5.14. The following conditions are equivalent:

(1) The space GS~ is T1. (2) The space GS~ is T0. (3) S = {0} or S is not a group. (4) S = {0} or Tx∈S(x + S)= ∅. Proof. Since the implications (1) ⇒ (2) ⇒ (3) are trivial, it remains to show that (3) ⇒ (4) ⇒ (1). (3) ⇒ (4) Assuming that (4) does not hold, we conclude that S 6= {0} and the intersection Tx∈S(x + S) contains some point y. Then y − S ⊆ S, y ∈ S + S = S and S − S = y − S + S ⊆ S + S = S, which means that S = S − S 6= {0} is a subgroup of G. But this contradicts (3).

(4) ⇒ (1) Assume that the condition (4) holds. To show that GS~ is a T1-space, ﬁx any non-zero element x ∈ G. If S = {0}, then S ∈ τG~ is a neighborhood of 0 that does not contain x. If Ty∈S (y +S)= ∅, then x∈ / y +S for some y ∈ S and {0}∪(y +S) ∈ τS~ is a neighborhood of 0 that does not contain x.

Proposition 5.15. The space GS~ is feebly compact if and only if the subgroup S − S has ﬁnite index in G.

Proof. If the space GS~ is feebly compact, then so is its continuous image GS. Now Proposi- tion 5.7 implies that the subgroup S −S has finite index in G. Now assume that the subgroup H = S − S has finite index in G and find a finite set F ⊆ G such that G = F + H. Let U be a locally finite family of nonempty open sets in G. Therefore for each x ∈ F we can pick sx ∈ S such that a family Ux = {U ∈ U : U ∩ (x + sx + S) 6= ∅} is finite. Let U ∈ U be any set. Since U ⊆ F + S − S, there exist x ∈ F and s,s′ ∈ H such that x + s − s′ ∈ U. Since the set U is open, there exists s′′ ∈ S such that x + s − s′ + s′′ + S ⊆ U. Then ′′ ′ ′′ x + sx + s + s ∈ (x + sx + S) ∩ (x + s − s + s + S), so U ∈Ux. Thus the family U = Sx∈F Ux is finite, which follows that the space GS~ is feebly compact. Proposition 5.16. The following conditions are equivalent: ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 29

(1) The space GS~ is compact. (2) The space GS~ is ω-bounded. (3) The space GS~ is totally countably compact. (4) The space GS~ is sequentially compact. (5) The space GS~ is countably compact. (6) The paratopological group GS~ is precompact. (7) The semigroup S is a subgroup of ﬁnite index in G.

Proof. The implications (1) ⇒ (2) ⇒ (3) are trivial. (3) ⇒ (7) Assuming that the space GS~ is totally countably compact, we conclude that its continuous image GS is totally countably compact, too. By Proposition 5.8, S is a subgroup of ﬁnite index in G. The implications (7) ⇒ (1) ⇒ (6) are trivial and (6) ⇒ (7) follows from Proposition 5.8. The implications

(7) ⇒ (4) ⇒ (5) are trivial. Finally we prove that (5) ⇒ (7). Assume that the space GG~ is countably compact and hence feebly compact. By Proposition 5.15, the subgroup S − S has finite index in G. Next, we check that S − S = S. Given any x ∈ S, we should prove that −x ∈ S. This is clear if x has finite order in G. So, assume that the order of x is infinite and hence the set X = {−nx : n ∈ N} is infinite. By the countable compactness, this set has an accumulation point a ∈ G. Then the neighborhood a + S ∈ τS~ of a contains some point −nx and hence −nx ∈ a + S and a ∈ −nx − S ⊆ −S. Since a ∈ −S is an accumulation point of the set X, the neighborhood {a}∪ S = {a}∪ (a + (−a)+ S) ∈ τS~ of a contains some point −mx 6= a with m ∈ N. Then −x = −mx + (m − 1)x ∈ S + S ∈ S and hence S is a subgroup of G. Proposition 5.17. The following conditions are equivalent:

(1) The space GS~ is countably pracompact. (2) The space GS~ is sequentially pracompact. (3) The subgroup S − S has finite index in G and S ⊆ C − S for some countable set C ⊆ S. Proof. The implication (2) ⇒ (1) is trivial. To prove that (1) ⇒ (3), assume that the space GS~ is countably pracompact and hence feebly compact. By Proposition 5.15, the subgroup H = S − S has finite index in G. If S is countable, then we can put C = S and conclude that S ⊆ C − S. So, assume that S is uncountable. The countable pracompactness of the paratopological group GS~ implies the countable pracompactness of its clopen subgroup HS~ . Consequently, HS~ is countably compact at some dense subset A ⊆ S − S. If A is finite, then choose any point b ∈ S \ A and observe that for every s ∈ S the basic neighborhood {b}∪(b+s+S) of b intersects A and hence s ∈−b+A−S ⊆ A−S. Since A ⊆ S −S is finite, there exists a finite set C ⊆ S such that A ⊆ C −S. Then S ⊆ A−S ⊆ C −S −S ⊆ C −S and we are done. Now consider the case of infinite set A. In this case we can choose a countable infinite subset B ⊆ A and using the countable compactness of HS~ at A, find an accumulation point b ∈ H of the set B. For every s ∈ S the basic neighborhood {b}∪ (b + s + S) of b has infinite intersection with the set B. Write b as b = b+ − b− for some b+, b− ∈ S and observe that the set s+S ⊃ b+ +s+S = b+s+b− +S has infinite intersection with B. Consequently, s ∈ B − S for every s ∈ S. Since B ⊆ A ⊆ S − S, we can find a countable set C ⊆ S such that B ⊆ C − S. Then S ⊆ B − S ⊆ C − S − S ⊆ C − S and we are done. (3) ⇒ (2) Assume that the condition (3) holds. Then the subgroup H = S − S has finite index in G and the space GS~ is homeomorphic to the product HS~ × F for some finite discrete space F . 30 TARAS BANAKH AND ALEX RAVSKY

The condition (3) yields a countable set C = {cn}n∈ω ⊆ S such that S ⊆ C − S and hence n H = S − S ⊆ C − S − S = C − S. Consider the countable set K := {0} ∪ {Pi=0 ci : n ∈ ω}. We claim that K is a sequence converging to zero and a dense set in HS~ . To see the latter it suﬃces to check that for any elements x,y,z ∈ S the set x − y + z + S intersects K. It follows from S − S ⊆ C − S that x − y + z ∈ cn − S for some n ∈ ω. Then n n−1 n−1 K ∋ X ci = cn + X ci ∈ (x − y + z + S)+ X ci ⊆ (x − y + z)+ S + S = x − y + z + S, i=0 i=0 i=0 n witnessing that K is dense in HS~ . To see that the sequence {Pi=0 ci}n∈ω converges to zero, take any s ∈ S and consider the basic neighborhood {0}∪ (s + S) ∈ τS~ of zero. It follows from s ∈ S ⊆ C − S that s ∈ ck − S for some k ∈ ω. Then for every m ≥ k we get m Pi=0 ci ∈ ck + S ⊆ s + S, which means that the neighborhood {0}∪ (s + S) contains all but ﬁnitely many points of the set K. Proposition 5.18. The following conditions are equivalent:

(1) The paratopological group GS~ is 2-pseudocompact. (2) Each power of GS~ is 2-pseudocompact. (3) The space GS~ is countably o-compact. (4) The subgroup S − S has finite index in G and each countable subset of S − S is contained in the set a − S for some a ∈ S. (5) Each infinite subset of G has infinite intersection with the set a − S for some a ∈ G. Proof. The implications (3) ⇒ (1) is trivial and (1) ⇒ (4) ⇒ (5) follows from Proposition 5.9 (and the preservation of 2-pseudocompactness by continuous homomorphisms of paratopological groups). (5) ⇒ (3) Assume that the condition (5) is satisfied. To prove that the space

GS~ is countably o-compact, fix a decreasing sequence (Un)n∈ω of nonempty open sets in GS~ . For every n ∈ ω fix a point un ∈ Un such that un + S ⊆ Un. By the condition (5), there exists a point a ∈ G such that the set Ω = {n ∈ ω : un ∈ a−S} is infinite. Then for every n ∈ Ω the point a belongs to the set un + S ⊆ Un and hence a ∈ Tn∈Ω Un = Tn∈ω Un, witnessing that the space GS~ is countably o-compact. Since (2) ⇒ (1), it remains to prove that (4) ⇒ (2). Given any cardinal κ, fix a decreasing sequence (Un)n∈ω of nonempty open sets in the space Gκ . For every n ∈ ω fix an element u ∈ U and find a finite set F ⊆ κ and a function S~ n n n κ sn ∈ S such that the basic open set κ Vn := \ {u ∈ GS~ : u(α) ∈ {un(α)}∪ (un(α)+ sn(α)+ S)} α∈Fn is contained in Un. Since the subgroup H = S − S has finite index in G, there exists a finite set D ⊆ G such that H + D = G and D ∩ (x + H)= {x} for every x ∈ D. For every n ∈ ω the κ κ κ element un ∈ G can be uniquely written as un = hn +fn for some hn ∈ H and fn ∈ D . By κ the condition (4), there exists a function a ∈ H such that {hn(α)+sn(α) : n ∈ ω}⊆ a(α)−S for every α ∈ κ. Then for every α ∈ κ we have

a(α)+ fn(α) ∈ hn(α)+ fn(α)+ sn(α)+ S = un(α)+ sn(α)+ S κ and hence a + fn ∈ Vn ⊂ Un for every n ∈ ω. Since the space D is compact, the sequence κ (fn)n∈ω has an accumulation point f ∈ D . We claim that −a − f ∈ Tn∈ω −Un. This will follow from the non-decreasing property of (Un)n∈ω as soon as we show that any open neighborhood W ⊆ GS~ of −a − f intersects inﬁnitely many sets −Un. For the neighborhood ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 31

κ κ W we can find a finite set E ⊆ κ and a function s ∈ S such that α∈E{w ∈ G : w(α) = T κ −a(α) − f(α)} ⊆ W . Since f is an accumulation point of the sequence (fn)n∈ω in D , the set Ω := Tα∈E{n ∈ ω : fn(α)= f(α)} is infinite. For every n ∈ Ω the point −a − fn belongs to W . On the other hand, a + fn ∈ Vn ⊆ Un and hence −a − fn ∈−Un. Propositions 5.16 and 5.18 imply the following characterization.

Corollary 5.19. If the semigroup S is countable, then the space GS~ is compact if and only if it is 2-pseudocompact.

Proof. The “if” part is trivial. Suppose that the group GS~ is 2-pseudocompact. It suﬃces to show that S is a group. By Proposition 5.18, S − S ⊆ a − S for some a ∈ S. In particular, 2a = a − a′ for some a′ ∈ S. Then a = −a′ ∈−S and so S − S ⊆ a − S ⊆−S − S ⊆−S.

Corollary 5.20. The space GS~ is compact if and only if the paratopological group GS~ is countably pracompact and 2-pseudocompact. Proof. The “only if” part is trivial. To prove the “if” part, assume that the paratopological group GS~ is countably pracompact and 2-pseudocompact. By Propositions 5.17 and5.18, the subgroup H = S − S has ﬁnite index in G, S ⊆ C − S for some countable subset C ⊆ S such that C ⊆ a − S for some a ∈ S. Then S ⊆ C − S ⊆ a − S − S = a − S and −S ⊆ S − S = −a + (S − S) = (−a + S) − S ⊆−S − S = −S, which implies that S = S − S is a subgroup of G. By Proposition 5.16, the space GS~ is compact.

Proposition 5.21. If space GS~ is Baire then either S is a group or S 6⊆ C − S for any countable set C ⊆ S. Proof. Suppose that S ⊆ C − S for some countable subset C ⊆ S. Then S − S ⊆ C − S − S = C − S. Observe that for every c ∈ C the set c − S is closed in GS~ . Since the subgroup S − S is open in the Baire space GS~ , there exists c ∈ C such that the closed set c − S has nonempty interior in S − S. Then a + S ⊆ c − S for some a ∈ S − S and hence (a − c)+ S ⊆−S. Then −S ⊆ S − S = (a − c)+ S − S ⊆−S − S = −S and hence S = S − S is a subgroup of G.

Corollary 5.22. The space GS~ is compact if and only if GS~ is countably pracompact and Baire.

Proof. If the space GS~ is compact, then by Proposition 5.16 , the semigroup S is a subgroup of finite index in G. The definition of the topology τS~ implies that each open dense subset of G coincides with G, which implies that the space GS~ is Baire. Being compact, the space GS~ is countably pracompact. Now assume that the space GS~ is countably pracompact and Baire. By Proposition 5.17, the subgroup S − S has finite index in G and S ⊆ C − S for some countable set C ⊆ S. By Proposition 5.21, S = S − S and by Proposition 5.16, the space GS~ is compact.

A space X is called a P -space if every Gδ-subset of X is open.

Proposition 5.23. The space GS~ is a P -space if and only if each countable subset C ⊆ S is contained in the set a − S for some element a ∈ S.

Proof. The characterization is trivial if S = {0}. In this case the space SS~ is discrete and hence a P -space. Also S ⊆ −S. So, we assume that S 6= {0}. Given any countable subset C ⊆ S, consider the Gδ-set W = Tc∈C({0}∪ (c + S)) in GS~ . If GS~ is a P -space, then W is a neighborhood of zero. Then there exists an element b ∈ S such that b+S ⊆ W . If b 6= 0, then 32 TARAS BANAKH AND ALEX RAVSKY put a := b. If b = 0, then take any point a ∈ S \{0} and conclude that a+S ⊆ S = b+S ⊆ W . In both cases there exists an element a ∈ S \ {0} such that a + S ⊆ W . Then for every c ∈ C the inclusion 0 6= a ∈ a + S ⊆ {0}∪ (c + S) implies a ∈ c + S and hence c ∈ a − S and ﬁnally C ⊆ a − S. Now assuming that each countable subset of S is contained in the set

a − S for some a ∈ S, we shall prove that GS~ is a P -space. Given any point x ∈ G and a sequence (Un)n∈ω of neighborhoods of x, we should prove that the intersection W := Tn∈ω Un is a neighborhood of x. For every n ∈ ω choose a point cn ∈ S such that x + cn + S ⊆ Un. By our assumption, there exists an element a ∈ S such that {cn : n ∈ ω} ⊂ a − S. Then x + a + S ⊆ x + cn + S ⊆ Un for each n ∈ ω. Hence x + a + S ⊆ Tn∈ω Un = W and thus W is a neighborhood of x in GS~ . Propositions 5.18, 5.15, 5.23 imply the following topological characterization of the 2- pseudocompactness of the paratopological group GS~ .

Corollary 5.24. The paratopological group GS~ is 2-pseudocompact if and only if GS~ is a feebly compact P-space. On the other hand, Corollaries 5.20 and 5.24 imply another characterization of the compactness of GS~ .

Corollary 5.25. The space GS~ is compact if and only if GS~ is a countably pracompact P- space. The following example shows that a counterpart of Proposition 5.11 does not hold for the paratopological groups GS~ . Example 5.26. For the group G = R and the subsemigroup S := [0, ∞) of G, the paratopological group GS~ is ω-precompact and not Lindel¨of. By Propositions 5.17 and 5.18, the paratopological group GS~ is countably pracompact but not 2-pseudocompact. Example 5.27. There are an Abelian group G and a subsemigroup S of the group G such that the paratopological groups GS and GS~ have the following properties: (1) GS is T0, sequentially compact, not totally countably compact, not precompact, and not a topological group; (2) GS~ is T1, 2-pseudocompact, not countably pracompact, not precompact, and not a topological group.

Proof. Let G = L Z be the direct sum of ω1 many copies of the group Z. In the group G α∈ω1 consider the subsemigroup

S = {0}∪ (xα)α∈ω1 ∈ G : ∃β ∈ ω1 (xβ > 0 and ∀α>β xα = 0) . Since S ∩ (−S) = {0}, we can apply Propositions 5.2, 5.14 and conclude that the space GS is T0 and the space GS~ is T1. Since G = S − S and for each countable inﬁnite subset C of S there is an element a ∈ S such that C ⊂ a − S, we see that by Proposition 5.9, the group GS is sequentially compact and by Proposition 5.18, the group GS~ is 2-pseudocompact. By Proposition 5.17, the group GS~ is not countably pracompact. By Proposition 5.8, the group GS is not totally countably compact and not precompact. Therefore the group GS~ is not precompact, too.

Remark 5.28. In [80], S´anchez proved that the group GS~ from Example 5.27 is a P -space. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 33

The following proposition was announced in [75].

Proposition 5.29. Each T1 paratopological group G with a countably compact square G × G is a topological group. Proof. Indeed, G∗ = {(x,x−1) ∈ G × G : x ∈ G} is a closed (and hence a countably compact) subgroup of G × G. Since G∗ has a base at the identity consisting of symmetric open neighborhoods, it is a topological group. By Lemma 3.8, the group G∗ is precompact. Since a projection π1 : G × G → F onto the ﬁrst coordinate is a continuous surjective homomorphism ∗ and π1(G ) = G, the group G is precompact too. By Proposition 3.15, G is a topological group.

Now we see that Proposition 5.29 cannot be generalized to T1 sequentially pracompact groups by Example 3.18 and to T0 sequentially compact groups by Example 5.27.

5.3. Cowide topologies. Topologies τ and σ on a set X are deﬁned to be cowide if for any nonempty sets U ∈ τ and V ∈ σ the intersection U ∩V is nonempty. If a topology σ is cowide to itself then the topology σ will be called wide. For two topologies τ and σ on a set X by τ ∨ σ we denote their supremum; it is generated by the base {U ∩ V : U ∈ τ, V ∈ σ}. τ∨σ τ Lemma 5.30. Let τ and σ be cowide topologies on a set X. Then W = W for each set τ∨σ τ W ∈ τ. Moreover, if the topology σ is wide, then W = W for each set W ∈ τ ∨ σ. τ∨σ τ Proof. The inclusion τ ∨ σ ⊇ τ implies the inclusion W ⊆ W for any subset W ⊆ X. τ Suppose that W ∈ τ, x ∈ W and U ∩ V is an arbitrary neighborhood of the point x such that U ∈ τ and V ∈ σ. Then W ∩ U is a nonempty τ-open set. Since the topologies τ τ∨σ and σ are cowide, the intersection W ∩ U ∩ V is nonempty. Therefore x ∈ W and hence τ τ∨σ τ W = W . Now suppose that the topology σ is wide, W ∈ τ ∨σ, x ∈ W and U ∩V ∈ τ ∨σ is any neighborhood of the point x such that U ∈ τ and V ∈ σ. The intersection W ∩ U is nonempty and belongs to the topology τ ∨ σ. Therefore there exist sets U ′ ∈ τ and V ′ ∈ σ such that ∅= 6 U ′ ∩ V ′ ⊆ W ∩ U. Since the topology σ is wide, the intersection V ∩ V ′ is nonempty. Since the topologies τ and σ are cowide, the intersection U ′ ∩ V ∩ V ′ ⊆ W ∩ U ∩ V τ∨σ is nonempty. Therefore x ∈ W .

5.3.1. Semiregularizations. Lemma 5.31. Let τ and σ be cowide topologies on a set X such that the topology σ is wide. Then (τ ∨ σ)sr = τsr.

Proof. The inclusion τ ∨ σ ⊇ τ implies the inclusion (τ ∨ σ)sr ⊇ τsr. So we show the opposite τ∨σ inclusion. Let W = intτ∨σ W be any regular open set in the topology τ ∨ σ. Lemma 5.30 τ∨σ τ τ τ implies that W = W . The inclusion τ ∨σ ⊇ τ implies the inclusion intτ∨σ W ⊇ intτ W . τ So we show the opposite inclusion. Let x ∈ intτ∨σ W be any point. Therefore there exist sets τ τ τ U ′ ∈ τ and V ′ ∈ σ such that x ∈ U ′ ∩ V ′ ⊆ W . Then U ′ ∩ V ′ ⊆ W . Since the topologies τ τ τ and σ are cowide, the set U ′ ∩ V ′ is τ-dense in the set U ′. Therefore U ′ ∩ V ′ = U ′ . Hence ′ ′τ τ τ x ∈ U ⊆ U ⊆ W and thus x ∈ intτ W .

Remark that similar constructions were considered by Bourbaki in order to provide non- regular topologies ﬁner than regular, see Exercise 20 for Chapter 1, §8 in [19]. 34 TARAS BANAKH AND ALEX RAVSKY

5.3.2. Feeble compactness. Lemmas 5.31 and 3.5 imply the following Proposition 5.32. Let τ and σ be cowide topologies on a set X such that the space (X,τ) is feebly compact and the topology σ is wide. Then the space (X,τ ∨ σ) is feebly compact, too. 5.3.3. H-compactness. We recall that a space X is H-compact iff for each open cover U of the space X there exists a finite subfamily V ⊆U such that X = SV ∈V V . It is easy to see that a regular H-compact space is compact and a space is H-compact if and only if its semiregularization is H-compact. This characterization combined with Lemma 5.31 imply the following proposition. Proposition 5.33. Let τ and σ be cowide topologies on a set X such that the space (X,τ) is H-compact and the topology σ is wide. Then the space (X,τ ∨ σ) is H-compact, too. Proposition 5.34. For any paratopological group (G, τ) the following conditions are equivalent: (1) The space (G, τ) is H-compact. (2) The space (G, τsr) is H-compact. (3) The space (G, τsr) is compact. If the space (G, τ) is quasiregular, then the conditions (1)–(3) are equivalent to (4) The space (G, τ) is compact. Proof. The implications (1) ⇔ (2) ⇐ (3) ⇐ (4) are trivial and hold for all spaces. To prove that (2) ⇒ (3), assume that the space (G, τsr) is H-compact. By [70, Ex. 1.9], [71, p. 31], and [71, p. 28], (G, τsr) is a regular paratopological group, so it is compact. Now assuming that the space (G, τ) is quasiregular, we shall prove that (3) ⇒ (4). Assume that the space (G, τsr) is compact. By Proposition 3.15, the quasiregular compact paratopological group (G, τsr) is a topological group. Then by Lemma 3.4, (G, τ) is a topological group, too. Therefore the space (G, τ) is regular and hence τsr = τ and the space (G, τ) = (G, τsr) is compact. 5.3.4. Applications of cowide topologies to the cone topologies. From now on and till the end of the section we assume that G is an Abelian group and S ∋ 0 is a subsemigroup in G. Let τ be any semigroup topology on G. It is easy to see that the cone topology τS defined at the beginning of Section 5 coincides with the supremum of the topologies of the spaces (G, τ) and GS. Similarly, the shift cone topology τS~ is the superemum of topologies of the spaces (G, τ) and GS~ . It is clear that the group G endowed with the topology τS~ is a paratopological group. Proposition 5.35. The following conditions are equivalent: (1) The topology of the space GS is wide. (2) The topology of the space GS~ is wide. (3) G = S − S.

Proof. Since the topology of the space GS is coarser than the topology of the space GS~ , we have implication (2) ⇒ (1). (1) ⇒ (3) Let x ∈ G be any element. If the topology of the space GS is wide, then the intersection S ∩ (x + S) of open subsets S and x + S of GS is nonempty and hence x ∈ S − S. (3) ⇒ (2) Observe that every nonempty open subset of the group GS~ contains the set x + S for some element x ∈ G. Let U1, U2 be any nonempty open subsets of the space GS~ . Then there are elements x1,x2 ∈ G such that x1 + S ⊆ U1 and x2 + S ⊂ U2. If G = S − S, then there are elements y1,y2, z1, z2 ∈ S such that x1 = y1 − z1 and x2 = y2 − z2. ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS 35

Then the intersection U1 ∩ U2 ⊃ (x1 + S) ∩ (x2 + S) contains the element y1 + y2 and hence is not empty. This means that the topology of the space GS~ is wide. Proposition 5.36. The following conditions are equivalent: (1) The topology of the space GS and the topology τ are cowide. (2) The topology of the space GS~ and the topology τ are cowide. (3) S is a dense subsemigroup of the group (G, τ).

Proof. Since the topology of the space GS is coarser than the topology of the space GS~ , we have implication (2) ⇒ (1). The implication (1) ⇒ (3) follows from the opennes of the set S is the paratopological group GS~ . The implication (3) ⇒ (2) follows from the observation that each nonempty open subset of the space GS~ contains a set x + S, x ∈ S, which is dense the space (G, τ). Proposition 5.37. Assume that the semigroup S is dense in (G, τ) and G = S − S. Then the following conditions are equivalent: (1) The space (G, τ) is feebly compact. (2) The space (G, τS) is feebly compact. (3) The space (G, τS~ ) is feebly compact. Proof. The implications (3) ⇒ (2) ⇒ (1) are obvious, and (1) ⇒ (3) follows from Proposi- tions 5.36, 5.35, and 5.33. Proposition 5.38. Assume that the semigroup S is dense in (G, τ) and G = S − S. Then the following conditions are equivalent: (1) The space (G, τ) is H-compact. (2) The space (G, τS) is H-compact. (3) The space (G, τS~ ) is H-compact. Proof. The implications (3) ⇒ (2) ⇒ (1) are obvious, and (1) ⇒ (3) follows from Proposi- tions 5.36, 5.35, and 5.33.

6. Acknowledgements The second author is grateful to Artur Tomita and Manuel Sanchis for providing him their articles. His sincere thanks also go to Minnan Normal University (Fujian, China) and to University of Hradec Kr´alov´e(Czech Republic) for the hospitality and opportunity to give lectures on the topic of the present paper, to Mikhail Tkachenko who has inspired him to prove Proposition 3.13, and to MathOverflow user Dog 69, who pointed him1 a reference from [19] related to Lemma 5.31.

1https://math.stackexchange.com/q/3138278 36 TARAS BANAKH AND ALEX RAVSKY

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Taras Banakh: Ivan Franko National University of Lviv (Ukraine), and Jan Kochanowski University in Kielce (Poland) E-mail address: [email protected]

Alex Ravsky: Department of Analysis, Geometry and Topology, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine, Naukova 3-b, Lviv, 79060, Ukraine E-mail address: [email protected]