On Finest Unitary Extensions of Topological Monoids

Topol. Algebra Appl. 2015; 3:1–10

Research Article Open Access

Boris G. Averbukh* On finest unitary extensions of topological monoids

Abstract: We prove that the Wyler completion of the unitary Cauchy space on a given Hausdorff topological monoid consisting of the underlying set of this monoid and of the family of unitary Cauchy filters on it, is a

T2-topological space and, in the commutative case, an abstract monoid containing the initial one. Keywords: topological monoid, Cauchy space, completion

MSC: 22A15, 54D35, 54E15

DOI 10.1515/taa-2015-0001 Received September 11, 2014; accepted November 11, 2014.

Introduction

For an arbitrary Hausdorff topological monoid, the concept of a unitary Cauchy filter (for brevity, wecallthem C-filters) which was defined in [1], generalizes the notion of a fundamental sequence of reals. Itwasproved in [1] that the underlying set X of this monoid X endowed with the family of such filters forms a Cauchy space called a unitary Cauchy space over X. Its convergence structure defines a T 1 -topology on X which is also 3 2 said to be unitary. In this paper, we begin the consideration of unitary completions of this monoid, i.e. such its extensions where all its C-filters converge. We use some principal ideas of the theory of Cauchy spaces but almost donot use its results since stronger statements are true in our case. However, we adduce all necessary definitions to make the paper as self-contained as it is possible. Each Cauchy space has many completions which have been studied in papers of many authors (see [2– 11]). The most important of them is perhaps the Wyler completion possessing the universal property over other completions. The proved in [1] existence of the least with respect to inclusion C-filters allows us to simplify its construction and proofs of its properties. In particular, we prove that the Wyler completion X˜ of the unitary

Cauchy space over X is a T2-topological space, and the function X → X˜ generates a covariant functor from the category of topological monoids and their homomorphisms into the category of T2-topological spaces. If X is commutative, then X˜ is an abstract monoid, and a canonical homeomorphic embedding i of X endowed with its unitary topology into X˜ is an algebraic embedding. However, the multiplication in X˜ can be discontinuous outside of (i × i)(X × X). The space X˜ possesses the universal property over other extensions of the set X to a topological space where all C-filters on X converge. Therefore, we call it the finest unitary extension of X. Like the unitary topology on X, the topology of X˜ is mostly rather fine, but all more suitable extensions of X possessing the above property, can be obtained from it by means of topological operations.

*Corresponding Author: Boris G. Averbukh: Moscow State Forestry University, department of mathematics, Moscow, Russian Federation E-mail: [email protected]

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1 The concept of a finest unitary extension

A) We begin this section with a quotation of main definitions and statements of paper [1]. Let X = (X, m, τ) be a Hausdorff topological monoid with an identity 1. Here m is a multiplication and τ a topology on the set X. In the following, we always shorten m(a, b) to ab. Similarly, we write sometimes X instead of (X, τ) for the underlying space.

Definition 1.1. A net S = {xα}α∈A in X is called a C-net (a left, a right C-net) if for each neighborhood U of 1 ′ there exists α0 ∈ A and for each α ≥ α0 there exists α0 ∈ A such that xα′ ∈ Uxα U (xα′ ∈ xα U, xα′ ∈ Uxα) for ′ ′ all α ≥ α0. Here and in the following, the line on top denotes the topological closure. Later we discuss also the ver- + sion of this definition without this line. If X is (R0, +) with the usual topology, then C-sequences are exactly all increasing fundamental ones. Definition 1.2. A filter F on X (i.e. in the power set P(X)) is called a C-filter (a left, a right C-filter) on X if the set MU = {x ∈ X : UxU ∈ F} (respectively, LU = {x ∈ X : xU ∈ F}, RU = {x ∈ X : Ux ∈ F}) belongs to F for every neighborhood U of 1.

We also write MU (LU , RU ) as MU (F) (respectively, LU (F), RU (F)) if we wish to show its dependence on F.

It is evident, every left (right) C-net is a C-net. The inclusions LU ⊂ MU and RU ⊂ MU hold for any filter F. Therefore each left (right) C- filter is a C-filter. If X is commutative, then these concepts coincide. As a rule, we only consider C-filters in the following. Left and right C-filters appear sometimes if we discuss properties of left and right translations. All statements proved for C-filters are also true (with obvious changes) forleft and right C-filters. A net S in X is a C-net if and only if the corresponding filter F(S) is a C-filter. A filter F on X is a C-filter if and only if the corresponding net S(F) is a C-net.

Definition 1.3. Let F1, F2 be C-filters. We write F1 ≥ F2 if MU (F2) ∈ F1 for every neighborhood U of 1. We set F1 ≈ F2 if both F1 ≥ F2 and F2 ≥ F1 are true. By Proposition 1.7 from [1], ≥ is a quasi-order relation, and ≈ is an equivalence relation on the set of C- filters. Similar relations for left (right) C-filters are denoted by ≥L, ≥R, ≈L, and ≈R. By Corollary 2.4 from [1], for each C-filter F, there exists a unique equivalent C-filter which properly contains no other C-filter. It is equal to the intersection ofall C-filters from the equivalence class of F and is denoted by Flst. Such C-filters are said to be least with respect to inclusion or shorter ⊂-least. It is proved in [1] (see Theorem 2.5.) that the set X endowed with the collection Σ of all C-filters is a Cauchy space. Recall that a Cauchy space (Y, S) is a couple consisting of a set Y and of a collection S of filters in the power set P(Y) which satisfies the following conditions: 1) For each point y ∈ Y, the ultrafilter y˙ consisting of all subsets containing y, belongs to S.

2) If F1, F2 are filters such that F1 ⊂ F2 and F1 ∈ S, then F2 ∈ S. 3) If F1, F2 are filters from S such that all intersections F1 ∩ F2 with F1 ∈ F1, F2 ∈ F2 are non-empty, then F1 ∩ F2 ∈ S. This definition was given by H. H. Keller [3]. It is a generalization of the concept of a uniform space which the notion of a completion still make sense for. The reader can find a detailed exposition of the theory of Cauchy spaces in [6] and [7].

Each Cauchy space (Y, S) defines a convergence structure qS on Y: a filter F ∈ S qS-converges to a point y ∈ Y iff the filter F∩y˙ belongs to S. The closure operator clqS of this structure is defined by the next condition: for any subset A ⊂ Y and for any point y ∈ Y, y belongs to clqS A iff there exists a filter F ∈ S such that A ∈ F and F qS-converges to y.

By Theorem 3.2 from [1], for Y = X, S = Σ, this operator defines a T 1 -topology on X. We call it the unitary 3 2 one. For any x ∈ X, the filter x˙ lst is the neighborhoods filter of x in this topology. If a C-filter F qΣ-converges to some x, then it converges to the same x in the unitary topology. If some filter F on X converges to a point x in the unitary topology, then it is a C-filter on X which qΣ-converges to x.

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B) Here, we will use some facts of the theory of completions of Cauchy spaces. The reader is referred to [2, 4, 5, 8–11] for its detailed exposition. A given Cauchy space is said to be complete if each its Cauchy filter converges in the convergence structure of this space. Let now (Y, S) be a Cauchy space, (Z, T) a complete Cauchy space, f : Y → Z an embedding such that clqT f (Y) = Z (clqT denotes the closure operator in (Z, T)) and f(F) ∈ T for any filter F ∈ S. Then (Z, T) is called a completion of (Y, S). Each Cauchy space has many completions. The most important one is perhaps the Wyler completion since it possesses the universal property over all other completions (see, for example, [2, 10, 11]). In the case of the unitary Cauchy space (X, Σ), the existence and the uniqueness of the ⊂- least C-filter in each equivalence class from Σ allows us to slightly simplify the known construction of the Wyler completion and a study of its properties. Denote by X˜ the set of all ⊂-least C-filters on X and by [F] the point of X˜ corresponding to a ⊂-least C-filter F. The injective map i : X → X˜ assigns to each point x the point i(x) = [x˙ lst]. Let Σ˜ denote the set of filters on X˜ consisting of all filters containing some filter of the form i(x˙ lst) or some filter of ˙ the form i(F) ∩ [F] where F is a qΣ-non-convergent (= non-convergent in the unitary topology on X) ⊂-least C-filter. Here, [F˙ ] is the trivial ultrafilter on X˜ corresponding to its point [F]. In particular, i(F) ∈ Σ˜ for any C-filter F, and [F˙ ] ∈ Σ˜ for any ⊂-least C-filter F. The last inclusion follows from the definition of Σ˜ if F does not converge in the unitary topology. If F converges in this topology, i.e. F = x˙ lst and [F] = i(x) for some x ∈ X, ˙ then it follows from the inclusion i(x) = i(x˙) ⊃ i(x˙ lst). The Wyler completion of (X, Σ) is the couple (X˜, Σ˜). The Wyler completion (X˜ l , Σ˜l) (respectively, (X˜ r , Σ˜r)) of the left (of the right) unitary Cauchy space can be constructed similarly by means of the set of ⊂-least left (respectively, right) C-filters. If the considered monoid X is commutative, then these three Wyler completions coincide. Return to the family Σ˜. For each ⊂-least C-filter F, if a filter G ∈ Σ˜ contains the filter i(F) or the filter ˙ i(F) ∩ [F], then the set i(F) ∪ {[F]} belongs to G for any F ∈ F. This set coincides with i(F) if F qΣ-converges, i.e. F = x˙ lst for some x ∈ X. It is showed at the beginning of section 2 B) of paper [1] that any different ⊂-least C-filters F1, F2 possess members F1 ∈ F1, F2 ∈ F2 with F1 ∩ F2 = ∅. The sets i(F1) ∪ {[F1]} and i(F2) ∪ {[F2]} do not intersect for such F1, F2. It implies that, for each filter G ∈ Σ˜, there exists an only ⊂-least C-filter F such that G contains the filter i(F) or the filter i(F) ∩ [F˙ ]. This filter F is said to be associated with G. For each ⊂-least C-filter F, the filter associated with [F˙ ] is F. For each filter F ∈ Σ, the filter associated with i(F) is its corresponding ⊂-least C-filters Flst. We prove now the following statement.

Proposition 1.4. The Wyler completion of the unitary Cauchy space is a complete T2 Cauchy space. Each filter ˜ G ∈ Σ qΣ˜ -converges to an only point, and this point is [F] where F is the ⊂-least C-filter associated with G. Proof. The first statement is a particular case of a well-known theorem, and we prove it only in order tomake the paper more self-contained. Show that (X˜, Σ˜) is a Cauchy space. Property 1) from the definition of a Cauchy space has already been proved. Property 2) follows from the definition of Σ˜.

Check property 3). Let G1, G2 ∈ Σ˜ and G1 ∩ G2 ≠ ∅ for any G1 ∈ G1, G2 ∈ G2. Denote by F1, F2 the ⊂-least C-filters associated with G1, respectively, G2. Any members of F1, F2 have a non-empty intersection. Hence, ˙ F1 = F2. Therefore, G1 ∩ G2 contains i(F1) or i(F1) ∩ [F1] and belongs to Σ˜. ′ ˜ ′ ˙′ It is now sufficient to verify the second statement. Indeed, for any [F ] ∈ X, G →qΣ˜ [F ] means that G∩[F ] ∈ ′ ′ ˙′ Σ˜. It is true for F = F. Let it be true for some F , and F1 be the ⊂-least C-filter associated with G ∩ [F ]. Then ˙′ ˙ both G and [F ] contain the filter i(F1) or the filter i(F1) ∩ [F1], i.e. F1 is associated with both the filters G and ˙′ ′ [F ]. Hence, F1 = F = F because of the uniqueness of the associated filter. ˜ Corollary 1.5. Each point ˜x ∈ X is the qΣ˜ -limit of a filter of the form i(F) where F ∈ Σ. If ˜x = i(x), then one of such filters is i(x˙ lst). If ˜x = [F] ∈ X˜ \ i(X), then it is the filter i(F) with the same F. For any C-filter F on X, the filter i(F) qΣ˜ -converges to an only point. For arbitrary equivalent C-filters F1 and F2 from Σ, the qΣ˜ -limits of the filters i(F1) and i(F2) coincide. Proof. The proof is evident. ˜ C) We prove in this section that the standard convergence structure qΣ˜ on X defines a Hausdorff topology.

Describe first the closure operator clqΣ˜ of this structure.

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Lemma 1.6. Let A be a subset of X˜ and A′ = i−1(A). Then: ′ (i) For any x ∈ X, i(x) ∈ clqΣ˜ A iff x ∈ clqΣ A . ˜ ′ (ii) For any [F] ∈ X \ i(X), [F] ∈ clqΣ˜ A iff either [F] ∈ A or A ∩ F ≠ ∅ for each F ∈ F. ˜ ˜ Proof. (i) Let ˜x be an arbitrary point of X. ˜x ∈ clqΣ˜ A means that there exists a filter G ∈ Σ such that A ∈ G and G →qΣ˜ ˜x. If ˜x = i(x), then, by the preceding Proposition, the ⊂-least C-filter associated with G is x˙ lst, i.e. ′ ′ i(x˙ lst) ⊂ G, and A ∩ i(H) ≠ ∅ for any H ∈ x˙ lst. It implies A ∩ H ≠ ∅ for all such H, and there exists a filter G ′ on X containing A and x˙ lst. By Proposition 2.2 from [1], it is a C-filter which is equivalent to x˙ lst. Hence, by ′ ′ ′ Proposition 2.3 from [1], G ∩ x˙ is a C-filter, G qΣ-converges to x, and x ∈ clqΣ A . To prove the converse, denote by G any filter on X˜ which contains A and all sets i(H) with H ∈ x˙ lst. Such a ′ ′ filter exists since x ∈ clqΣ A implies that there exists a C-filter F containing A and equivalent to x˙. This filter ′ contains x˙ lst, and A ∩ H ≠ ∅ for any H ∈ x˙ lst. Therefore, A ∩ i(H) ≠ ∅ for these H. The inclusion G ⊃ i(x˙ lst) ˜ holds for this filter G, and so G ∈ Σ. Hence, G →qΣ˜ i(x) by the previous Corollary. ˜ ˙ (ii) Let now ˜x = [F] ∈ X \ i(X). G →qΣ˜ [F] means that F is the associated with G filter and G ⊃ i(F) ∩ [F]. Therefore, A has non-empty intersections with all sets of the form i(F) ∪ {[F]} where F ∈ F. If [F] does not belong to A, then A′ ∩ F ≠ ∅ for all F ∈ F. To prove the converse, take as G any filter on X˜ which contains A ˙ ˜ and all sets i(F) ∪ {[F]} with F ∈ F. Then G ⊃ i(F) ∩ [F], G ∈ Σ and G →qΣ˜ [F] by Proposition 1.4. ˜ −1 −1 Remark 1.7. For any A ⊂ X, B ⊂ X, the statement (i) implies the equalities i (clqΣ˜ A) = clqΣ i (A) and i(clqΣ B) = i(X) ∩ clqΣ˜ i(B). Lemma 1.8. Each ⊂-least C-filter on X has a base consisting of open in the unitary topology sets and a base consisting of closed in this topology sets. Proof. Let F be a ⊂-least C-filter and A ∈ F. First, we prove that there exist a set B ∈ F and a neighborhood

V of 1 in the initial topology τ such that ∆V (B) = {x ∈ VBV : VxV ∩ B ≠ ∅} ⊂ A. By Lemma 2.10 from [1], we may assume that A belongs to the base of F consisting of all finite intersections of sets of the form MU (F) ′ and UxU where U runs neighborhoods of 1 in the topology τ and, for each U, x ∈ MU′ (F) with U ≺ U. (This relation means that there exists a neighborhood O of 1 in the initial topology such that OU′O ⊂ U.) Observe that U1 ⊂ U2 implies MU1 (F) ⊂ MU2 (F). Hence, we may only consider the intersections which contain an ′ only set of the form MU (F). Thus, let A = MU (F) ∩ U1x1U1 ∩ ··· ∩ Un xn Un where xi ∈ M ′ (F) with U ≺ Ui for Ui i i = 1, n. Take an arbitrary neighborhood U′ ≺ U and choose a neighborhood V of 1 (in the topology τ) such ′ ′ ′ ′ ′ ′ that VU V ⊂ U, VUi V ⊂ Ui for i = 1, n. Set B = MU′ (F) ∩ U1x1U1 ∩ ··· ∩ Un xn Un. Then it is easy to show that VBV ⊂ U1x1U1 ∩ ··· ∩ Un xn Un. For an arbitrary x ∈ X, VxV ∩ B ≠ ∅ implies VxV ∩ MU′ (F) ≠ ∅, and there ′ ′ exists y ∈ VxV such that U yU ∈ F. Therefore, UxU ∈ F, x ∈ MU (F) and {x : VxV ∩ B ≠ ∅} ⊂ MU (F). We have now ∆V (B) ⊂ MU (F) ∩ U1x1U1 ∩ ··· ∩ Un xn Un = A.

Show now that clqΣ B lies in the interior of A in the unitary topology. It is evident that ∆V (x) ⊂ A for each x ∈ B. Let z be an arbitrary point from clqΣ (X \ A) and O a neighborhood of 1 in the initial topology τ on X 2 such that O ⊂ V. If ∆O(z) ∩ B ≠ ∅ and x belongs to this intersection, then z ∈ ∆O(x) and ∆O(z) ⊂ ∆O(∆O(x)) ⊂ ∆V (x) ⊂ A by statements (iii) and (v) of Lemma 3.1. from [1]. But ∆O(z) ∩ (X \ A) ≠ ∅ by statement (i) of theorem

3.2 from [1]. This contradiction means that ∆O(z) ∩ B = ∅ and z ∉ clqΣ B, i.e. clqΣ (X \ A) ∩ clqΣ B = ∅. Thus, the interiors of all sets A ∈ F and the closures of all sets B ∈ F (in both these cases, in the unitary topology) form bases of F. It completes the proof. ˜ Theorem 1.9. (i) The operator clqΣ˜ defines a topology on X. (We call it natural and denote by τ˜(X) or simply by τ˜ when no confusion seems likely.) (ii) In the unitary topology on X and the natural topology on X˜, the map i : X → X˜ is a homeomorphism onto a dense open subspace, and the subspace X˜ \ i(X) is discrete. ˜ ˜ (iii) For any ˜x ∈ X, if a filter G ∈ Σ qΣ˜ - converges to ˜x, then it converges to ˜x in the natural topology. If some filter G on X˜ converges in the natural topology, then it belongs to Σ˜. (iv) The natural topology is Hausdorff. ˜ ′ −1 Proof. (i) Check axioms of the closure operator of a topological space. First, let A1, A2 ⊂ X, A1 = i (A1), ′ −1 ′ ′ ′ ′ A2 = i (A2), and ˜x ∈ clqΣ˜ (A1 ∪A2). If ˜x = i(x), then x ∈ clqΣ (A1 ∪A2) = clqΣ A1 ∪clqΣ A2 and ˜x ∈ clqΣ˜ A1 ∪clqΣ˜ A2 ˜ by Lemma 1.6 and since the operator clqΣ defines a topology on X by Theorem 3.2 from [1]. If ˜x = [F] ∈ X \ i(X)

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′ ′ ′ does not belong to A1 ∪ A2, then (A1 ∪ A2) ∩ F ≠ ∅ for any F ∈ F. It implies A1 ∩ F ≠ ∅ for each F ∈ F or ′ A2 ∩ F ≠ ∅ for each F ∈ F, i.e. ˜x ∈ clqΣ˜ A1 ∪ clqΣ˜ A2. 2 Check now the axiom (clqΣ˜ ) = clqΣ˜ . Let ˜x ∈ clqΣ˜ clqΣ˜ A. Consider two cases. First, let ˜x = i(x). Then −1 ′ ′ x ∈ clqΣ i (clqΣ˜ A) by statement (i) of Lemma 1.6. It implies x ∈ clqΣ (clqΣ A ) = clqΣ A and ˜x ∈ clqΣ˜ A. Let now ˜x = [F] where F is a divergent (in the unitary topology) ⊂-least C-filter on X. We assume [F] ∉ A, otherwise −1 ′ the statement is trivial. By Lemma 1.6, [F] ∈ clqΣ˜ clqΣ˜ A involves i (clqΣ˜ A) ∩ F = clqΣ A ∩ F ≠ ∅ for each F ∈ F. By Lemma 1.8, F has a base consisting of open in the unitary topology sets. Therefore, A′ ∩ F ≠ ∅ for all F ∈ F.

It means [F] ∈ clqΣ˜ A. Thus, clqΣ˜ clqΣ˜ A ⊂ clqΣ˜ A. The converse inclusion is trivial.

(ii) The map i : X → i(X) is one-to-one, and i(clqΣ B) = clqΣ˜ i(B) ∩ i(X) for any B ⊂ X. Hence, i is a ˜ ˜ homeomorphism onto the subspace i(X). By Lemma 1.6, clqΣ˜ A = A for each A ⊂ X \ i(X). Therefore, X \ i(X) is ˜ closed and discrete in the natural topology. Moreover, [F] ∈ clqΣ˜ i(X) for each [F] ∈ X \ i(X), i.e. i(X) is dense in X˜.

(iii) First, let ˜x = i(x). G →qΣ˜ i(x) implies G ⊃ i(x˙ lst). By statement (iii) of Theorem 3.2 from [1], the filter x˙ lst is generated by all neighborhoods of x in the unitary topology. Then, by statement (ii) of this theorem, the filter i(x˙ lst) contains all neighborhoods of i(x) in the natural topology on X˜. Therefore, it is true for G, too. To consider the case ˜x = [F] ∈ X˜ \ i(X), we need the next lemma. Lemma 1.10. Let [F] be an arbitrary point of X˜ \ i(X). A base of the natural topology at this point consists of sets of the form i(U′) ∪ {[F]} where U′ runs open (in the unitary topology) sets from F. Proof. Since the subspace X˜ \ i(X) is discrete, a small enough neighborhood U of [F] does not contain any points from this subspace except for [F]. The subset U′ = i−1(U) ⊂ X is open. Show that it belongs to F. If it ′ ˜ is false, then (X \ U ) ∩ F ≠ ∅ for any F ∈ F, and it implies by Lemma 1.6 that [F] ∈ clqΣ˜ (X \ U). However, it is impossible since U is a neighborhood of [F]. Conversely, prove that if U′ ⊂ X is open in the unitary topology and belongs to F, then U = i(U′) ∪ {[F]} is a neighborhood of [F]. We only need to verify that [F] lies inside −1 ˜ ′ ′ ˜ of U. Indeed, i (X \ U) = X \ U does not intersect U ∈ F. Hence, [F] ∉ clqΣ˜ (X \ U) by Lemma 1.6 (ii). ˙ Now, we return to the proof of Theorem 1.9. Let G →qΣ˜ [F]. It implies G ⊃ i(F) ∩ [F]. Hence, G contains all neighborhoods of [F] and converges to [F] in the natural topology. Prove the second statement from (iii). For an arbitrary filter G on X˜, let ˜x belong to lim G in the natural topology. Then G contains all neighborhoods of ˜x . If ˜x = i(x), then G ⊃ i(x˙ lst) since x˙ lst is generated by neighborhoods of x in the unitary topology (see Theorem 3.2 from [1]). It implies that G ∈ Σ˜. In the case ˜x = [F] ∈ X˜ \ i(X), G contains all sets of the form i(U′) ∪ {[F]} where U′ runs open (in the unitary topology) sets from F. Then G ⊃ i(F) ∩ [F˙ ] by Lemma 1.8., and G ∈ Σ˜.

(iv) Let ˜x1 = [F1], ˜x2 = [F2] be arbitrary different points of X˜. The ⊂-least C-filters F1 and F2 have bases ′ ′ consisting of open in the unitary topology sets. Therefore, there exist open in this topology U1 ∈ F1, U2 ∈ F2 ′ ′ ′ such that U1 ∩ U2 = ∅. Set U1 = i(U1) if F1 converges in the unitary topology, i.e. ˜x1 = i(x) for some x. In this ′ ′ case, x ∈ U1 implies ˜x1 ∈ U1. If F1 does not converges in this topology, then we set U1 = i(U1) ∪ {[F1]} and ˜x1 ∈ U1 is true again. Define similarly a set U2 for F2 and observe that U1, U2 are neighborhoods of ˜x1, ˜x2 in the natural topology with the empty intersection.The proof is complete. Definition 1.11. We call the set X˜ endowed with its natural topology τ˜ the finest unitary extension of the monoid X. Example 1.12. (i) Find the finest unitary extension of the monoid (Q, +) with the usual initial topology. For any topological group, its unitary topology coincides with the initial one, all C-filters are Cauchy filters of the Rölke (in the commutative case, of the standard) uniformity and ⊂- least C-filters are minimal ones. Therefore, we can associate each divergent ⊂-least C-filter with an irrational and assume that the set Q˜ consists of all points (x, 0) ∈ R2 with rational x and of all points (x, 1) ∈ R2 with irrational x. A base of the natural topology at the point (x, 0) consists of intervals Iϵ(x) = {(y, 0): y is rational and x − ϵ < y < x + ϵ} where ϵ > 0.A base at each point (x, 1) consists of sets of the form {(x, 1)} ∪ Iϵ(x). This topology is non-T3. The addition in Q˜ is only continuous at points with rational co-ordinates. + (ii) Let now X be (R0, +) with the usual topology. Two ⊂-least C-filters correspond to each x ≠ 0: the trivial ultrafilter x˙ and the filter Fx with the base consisting of intervals ]x − ϵ; x[ with ϵ > 0. The latter filters diverge in the unitary topology since this topology is discrete. As above, we can assume that the set X˜ consists of + + points of the form (x, 0) corresponding to the filters x˙ with x ∈ R0 and of points of the form (x, 1) with x ∈ R

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corresponding to the filters Fx. The natural topology is discrete on both these half-lines. By Lemma 1.10, its base at the point (x, 1) consists of the sets (x, 1) ∪ {(y, 0): x − ϵ < y < x} with 0 < ϵ < x. Remark 1.13. We prove in the next section that the finest unitary extension X˜ possesses the universal property over other extensions of X to a topological space where all C-filters of X converge. It is clear that the natural topology should mostly be rather fine for that. However, extensions with this property and with more interesting topologies (in particular, being topological monoids) can be obtained from X˜ by means of suitable topological operations. We do that in the following papers of this series.

2 Some properties of the finest unitary extension

A) In this section, we will use a few more properties of C-filters. Let Y be a Hausdorff topological monoid on a set Y and f a continuous identity preserving homomorphism of X into Y.

Proposition 2.1. (i) For each C-filter F on X, f (F) is a C-filter on Y. If F1 and F2 are C-filters on X such that F1 ≥ F2 (respectively, F1 ≈ F2) takes place, then f (F1) ≥ f(F2) (respectively, f(F1) ≈ f(F2)). (ii) Let now f be a dense algebraic and topological embedding and F an arbitrary filter on X. If f (F) is a

C-filter on Y, then F is a C-filter on X. If F1, F2 are filters on X such that f(F1), f(F2) are C-filters on Y, then f(F1) ≥ f (F2) (respectively, f(F1) ≈ f(F2)) implies F1 ≥ F2 (respectively, F1 ≈ F2). Proof. (i) Denote by 1X and 1Y the identities of X and of Y, respectively. Let V be an arbitrary neighborhood of 1Y and U a neighborhood of 1X such that f(U) ⊂ V. For each x ∈ MU (F), we have f(UxU) ⊂ f (U)f(x)f(U) ⊂ Vf(x)V. (For brevity, we denote equally the closure operators in both the monoids since it is always clear which monoid we consider.) Therefore, f(x) ∈ MV (f (F)). Hence, MV (f (F)) ⊃ f(MU (F)), MV (f (F)) ∈ f (F), and f(F) is a C- filter on Y.

In the case F1 ≥ F2, we have MU (F2) ∈ F1. Therefore, MV (f(F2)) ⊃ f(MU (F2)) ∈ f (F1) and f(F1) ≥ f (F2). (ii) Let now U be an arbitrary neighborhood of 1X and V a neighborhood of 1Y such that V ∩ f (X) = f (U). Then V = f(U). The inclusion MV (f(F)) ∈ f (F) implies that there exists F ∈ F such that f(F) ⊂ MV (f(F)). Vf(x)V = f(U)f(x)f (U) = f (UxU) ∈ f(F) is true for any x ∈ F and, therefore, f(UxU) = f (UxU) ∩ f(X) ∈ f(F), too. Hence, UxU ∈ F, and x ∈ MU (F) is true for x ∈ F, i.e. F ⊂ MU (F), MU (F) ∈ F, and F is a C- filter on X. To prove the second statement, denote by W an arbitrary neighborhood of 1Y such that W ≺ V. It means that there exists a neighborhood O of 1Y such that OWO ⊂ V. Let F ∈ F1 be a set such that f(F) ⊂ MW (f(F1)). Then f (F) ⊂ MV (f(F2)) and UxU ∈ F2 for any x ∈ F as above. It implies F ⊂ MU (F2), MU (F2) ∈ F1, and F1 ≥ F2. It follows from the proved statement that the function X → (X, Σ) generates a covariant functor from the category of Hausdorff topological monoids and their continuous identity preserving homomorphisms into the category of T2 Cauchy spaces. Corollary 2.2. (i) Let X′ be a submonoid of X and j the corresponding embedding. For an arbitrary C-filter ′ ′ F on X , j(F) is a C-filter on X. If F1, F2 are C-filters on X such that F1 ≥ F2 (respectively, F1 ≈ F2), then j(F1) ≥ j(F2) (respectively, j(F1) ≈ j(F2)). (ii) Let τ′ be a finer than τ topology on X such that X′ = (X, m, τ′) is a Hausdorff topological monoid, too. ′ ′ Then each C-filter on X is a C-filter on X. Moreover, F1 ≥ F2 (respectively, F1 ≈ F2) on X implies F1 ≥ F2 ′ (respectively, F1 ≈ F2) on X for any C-filters F1 and F2 on X . For an arbitrary element a ∈ X, let λa denote the corresponding left translation x → ax of X.

Proposition 2.3. For each left C-filter F, λa(F) is also a left C-filter. If F1 and F2 are left C- filters with F1≥LF2 (respectively, F1≈LF2), then λa(F1)≥L λa(F2) (respectively, λa(F1)≈L λa(F2)). A similar statement is true for right C-filters and right translations.

Proof. For any x ∈ X and for any neighborhood U of 1, the inclusion xU ∈ F implies axU ⊃ axU ∈ λa(F).

Therefore, LU (λa(F)) ⊃ λa(LU (F)) and LU (λa(F)) ∈ λa(F). In the case F1 ≥ F2, we have LU (F2) ∈ F1. Hence, LU (λa(F2)) ⊃ λa(LU (F2)) ∈ λa(F1) and λa(F1) ≥ λa(F2).

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Let now {Xs}s∈S be a family of Hausdorff topological monoids. For each s ∈ S, denote by Xs the under- ∏︀ lying space of Xs and by 1s its identity. Set X = s∈S Xs, and let ps be the canonical projection of X onto Xs. ∏︀ ∏︀ For a given family {Fs}s∈S, where Fs is a filter on Xs, denote by F = s∈S Fs the filter on X = s∈S Xs which ∏︀ is generated by all subsets of the form s∈S Fs with Fs ∈ Fs for any s. Proposition 2.4. i) F is a C-filter on X iff Fs is a C-filter on Xs for any s ∈ S. ′ ∏︀ ′ ∏︀ ′ ′ ii) For each s ∈ S, let Fs, Fs be C-filters on Xs and F = s∈S Fs, F = s∈S Fs. Then F ≥ F (respectively, ′ ′ ′ F ≈ F ) is true iff Fs ≥ Fs (respectively, Fs ≈ Fs) for any s ∈ S. Proof. i) If F is a C-filter, then Fs = ps(F), s ∈ S, is a C-filter by Proposition 2.1. Prove the converse. Let U be an arbitrary neighborhood of 1 = {1s}s∈S. There exists a family {Us}s∈S where Us is a neighborhood of ∏︀ 1s and Us ≠ Xs for a finite set of indexes, such that s∈S Us ⊂ U. Let x = {xs}s∈S be an arbitrary point ∏︀ ∏︀ ∏︀ of X such that xs ∈ MUs (Fs) for any s ∈ S. Then s∈S Us xs Us ∈ F and s∈S Us xs Us = s∈S Us xs Us = ∏︀ ∏︀ ∏︀ ( s∈S Us) x ( s∈S Us) ⊂ UxU. Therefore, UxU ∈ F and s∈S MUs (Fs) ⊂ MU (F). Hence, MU (F) ∈ F and F is a C-filter. ′ ′ ∏︀ ′ ′ ′ ii) Inclusions MUs (Fs) ∈ Fs and MU (F ) ⊃ s∈S MUs (Fs) imply MU (F ) ∈ F, i.e. F ≥ F . The converse ′ ′ follows from Proposition 2.1 since Fs = ps(F), Fs = ps(F ). Corollary 2.5. Let the family {Xs}s∈S be finite and F an arbitrary filter on X. Then: i) F is a C-filter iff ps(F) is a C-filter on Xs for any s ∈ S. ′ ′ ′ ′ ii) For any C-filters F, F on X, F ≥ F (respectively, F ≈ F ) holds iff ps(F) ≥ ps(Fs) (respectively, ps(F) ≈ ′ ps(Fs)) is true for any s ∈ S. Proof. i) If F is a C-filter, then all ps(F), s ∈ S, are C-filters by Proposition 2.1. Let now all ps(F), s ∈ S, be C- −1 ∏︀ filters. For each s ∈ S, take an arbitrary member Fs of ps(F). The set ps (Fs) belongs to F. Therefore, s∈S Fs = ⋂︀ −1 ∏︀ s∈S ps (Fs) ∈ F. It means F ⊃ s∈S ps(F). It follows from the first part of the preceding proposition that ∏︀ this product is a C-filter. Hence, by Proposition 2.2 from [1], F is a C-filter which is equivalent to s∈S ps(F). ∏︀ ′ ∏︀ ′ ii) In view of equivalences F ≈ s∈S ps(F), F ≈ s∈S ps(F ), the considered statement follows immediately from the second part of the preceding proposition.

We now turn our attention to the initial monoid X again. Let F1, F2 be filters on X. Denote by F1F2 the filter which is generated by all sets of theform F1F2 with F1 ∈ F1, F2 ∈ F2. It is evident, this operation is associative. It is commutative if X is commutative.

Corollary 2.6. Let X be commutative and F1, F2 C-filters on X. Then: (i) F1F2 is a C-filter, too. ′ ′ ′ ′ ′ ′ (ii) If F1, F2 are C-filters on X such that F1 ≥ F1, F2 ≥ F2 (respectively, F1 ≈ F1, F2 ≈ F2), then F1F2 ≥ ′ ′ ′ ′ F1F2, (respectively, F1F2 ≈ F1F2). Proof. Denote by f the map (a1, a2) → a1a2 : X × X → X. Since X is commutative, this map is a continuous identity preserving homomorphism. Moreover, f (F1 × F2) = F1F2, and the statement follows immediately from Propositions 2.1 and 2.4. If the underlying set X of a given Cauchy space (X, S) is endowed with an associative multiplication and

F1F2 ∈ S for any filters F1, F2 from S, then this triplet is called a semigroup Cauchy space (see [9]). Thus, if X is commutative, then (X, Σ, m) is a semigroup Cauchy space. B) We now prove that the function which assigns to each Hausdorff topological monoid its finest unitary extension, generates a covariant functor from the category of such monoids into the category of T2 topological spaces.

Let X, Y be Hausdorff topological monoids with the underlying spaces (X, τX) and (Y, τY ) and the finest unitary extensions X˜ and Y˜. Denote by iX and iY the canonical embeddings of X and Y into X˜ and Y˜, respectively.

Proposition 2.7. (i) For each continuous (in the topologies τX and τY ) identity preserving homomorphism f : X → Y, there exists a unique continuous (in the natural topologies on X˜ and Y˜) map ˜f : X˜ → Y˜ such that

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iX iY ˜f X˜ / Y˜ commutes. (ii) Denote by P the function from the class of continuous identity preserving homomorphisms of Haus- dorff topological monoids to the class of continuous mapsof T2 topological spaces which assigns to each such homomorphism f : X → Y of monoids the corresponding map ˜f : X˜ → Y˜ of their finest unitary extensions. This function P is a covariant functor from the category of Hausdorff topological monoids and their continuous identity preserving homomorphisms to the category of T2 topological spaces and their continuous maps. Proof. In order to define the map ˜f, consider an arbitrary point ˜x = [F] ∈ X˜. By Proposition 2.1, the filter f(F) is a C-filter on Y. Denote by ˜y the point of Y˜ corresponding to the ⊂-least C-filter on Y which is equivalent to ˜ f(F), and set f(˜x) = ˜y. If ˜x = iX(x) and y = f(x), then F = x˙ lst converges to x in the unitary topology on X. Therefore, f (F) converges to y in the unitary topology on Y since f is continuous in the pair of the unitary topologies on X and Y by Proposition 3.5 from [1]. Hence, f(F) ≈ y˙ and ˜y = [y˙ lst] = iY (y). It means that the preceding diagram commutes.

Remark 2.8. Our construction of the map ˜f is a special case of a more general one. Let (Z1, S1), (Z2, S2) be T2-Cauchy spaces and j1, j2 their canonical embeddings into corresponding Wyler completions (Z˜1, S˜1), (Z˜2, S˜2). It is well-known (see for example [2, 8, 10, 11]) that, for any Cauchy map g :(Z1, S1) → (Z2, S2), i.e. a map of the set Z1 into Z2 taking any Cauchy filter from S1 into a Cauchy filter from S2, there exists a canonical Cauchy map g˜ :(Z˜1, S˜1) → (Z˜2, S˜2) such that g˜ ∘ j1 = j2 ∘ g. Moreover, the function g → g˜ is a covariant functor in the category of T2-Cauchy spaces and their Cauchy maps. Show now that ˜f is continuous. By Proposition 2.1, g = f is a Cauchy map of the unitary Cauchy space on

X into the unitary Cauchy space on Y. Denote by Σ˜X and Σ˜Y the collections of filters on X˜ and Y˜, respectively, such as in the above definition of the Wyler completion of a unitary Cauchy space. In this particular case,the known construction of the map g˜ leads to the above map ˜f . Therefore, according to the previous Remark, the map ˜f is a Cauchy map between the Cauchy spaces (X˜, Σ˜X) and (Y˜ , Σ˜Y), i.e. ˜f(G) ∈ Σ˜Y whenever G ∈ Σ˜X. It is also not difficult to check immediately. Let now G be an arbitrary filter on X˜ converging in the natural topology.

By statement (iii) of Theorem 1.9, it belongs to Σ˜X. Then the filter ˜f(G) converges in the natural topology on Y˜ by Proposition 1.4. and Theorem 1.9 (iii). Since X˜ and Y˜ endowed with their natural topologies are Hausdorff spaces (see statement (iv) of Theorem 1.9), it means that ˜f is continuous in the natural topologies on X˜ and Y˜. The map ˜f is the unique extension of f with the required properties since each other one coincides with

˜f on the dense subset iX(X) ⊂ X˜. Check now that P preserves compositions of maps. Let Z be one more Hausdorff topological monoid and g : Y → Z a continuous identity preserving homomorphism. Denote by Z the underlying space, by Z˜ the finest unitary extension of Z and by iZ the corresponding canonical injection. For an arbitrary ˜x = [F] ∈ X˜, let G be the corresponding to P(f )(˜x) ⊂-least C-filter on Y which is equivalent to f(F). The point P(g) ∘ P(f)(˜x) corresponds to the ⊂-least C-filter on Z which is equivalent to g(G). The point P(g ∘ f)(˜x) corresponds to the ⊂-least C-filter on Z which is equivalent to (g ∘ f)(F) = g(f (F)). But these filters coincide since g(G) ≈ g(f(F)) by Proposition 2.1. The proof is complete. ˜ C) In this section, we study some algebraic structures on X˜. Let us first define an action λa : X˜ l → X˜ l of an arbitrary a ∈ X on the left finest unitary extension. Consider an arbitrary point ˜x = [F] ∈ X˜ l. By Proposition 2.3, the filter λa(F) which is generated by sets of the form aF with F ∈ F, is also a left C-filter. Then the point ˜λa(˜x) corresponds to the ⊂-least left C-filter which is equivalent to λa(F). An action ρ˜a of a ∈ X on the right finest unitary extension X˜ r can be defined similarly. ˜ ˜ ˜ ˜ It is evident that λab = λa ∘ λb for any a, b ∈ X. Moreover, i ∘ λa = λa ∘ i. Indeed, by Proposition 2.3 ˙ ˙ above and Proposition 2.2 from [1], for any x, a ∈ X, λa(x˙) ⊂ (ax) implies λa(x˙ lst) ≈ λa(x˙) ≈ (ax). Therefore, ˜ ˙ λa(i(x)) = (λa(x˙ lst))lst = (ax)lst = i(λa(x)). Proposition 2.9. For each a ∈ X, ˜λa and ρ˜a are continuous maps.

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Proof. Let [F], [G] be arbitrary points of X˜ such that ˜λa([F]) = [G], and U an arbitrary open in the unitary topology on X member of G. If [G] ∈ i(X), i.e. G = x˙ lst for some x ∈ X, then x ∈ U since x˙ lst is the neighborhoods filter of this point x in the unitary topology. By Lemma 1.10 and Theorem 1.9 (ii), if U runs all sets with required above property, then, for any [G] ∈ i(X) and for any [G] ∈ (X˜ \ i(X)), the set i(U) ∪ {[G]} runs a base of X˜ at the point [G].

By Proposition 3.8 from [1], the map λa is continuous in the unitary topology. Therefore, the inclusion −1 G ⊂ λa(F) implies that V = λa (U) is an open in the unitary topology member of F, and i(V) ∪ {[F]} is a neighborhood of [F]. For this neighborhood, we obtain the next continued equality: ˜λa(i(V) ∪ {[F]}) = ˜λa(i(V)) ∪ {˜λa([F])} = i(λa(V)) ∪ {[G]} = i(U) ∪ {[G]}, and the proof is complete. We consider now the case if X is commutative. Let ˜x = [F], ˜y = [G] be arbitrary points of X˜. By Corollary

2.6, FG is a C-filter, and we denote by (FG)lst the ⊂-least C-filter corresponding to it. Set now ˜x • ˜y = [(FG)lst]. Proposition 2.10. Let X be a commutative Hausdorff topological monoid. Then: (i) The set X˜ endowed with the multiplication • is a commutative monoid. (ii) The map i is an isomorphism of the monoid (X, m) onto a submonoid in (X˜, •). For an arbitrary a ∈ X, the translation of (X˜, •) about i(a) coincides with the action ˜λa = ρ˜a of a in X˜.

(iii) For any continuous (in their initial topologies τX and τY ) identity preserving homomorphism f : X → Y between commutative Hausdorff topological monoids, the map ˜f : X˜ → Y˜ which was defined in the proof of Proposition 2.7 above, is a homomorphism between abstract monoids (X˜, •) and (Y˜ , •). (We denote by • the above multiplications in both the sets X˜ and Y˜ and hope no confusion will arise for the reader because of it.) The function f → ˜f is a covariant functor from the category of commutative Hausdorff topological monoids into the category of commutative abstract monoids. Proof. In this proof, we will denote the identity of X by e. (i) It follows from the definition of the multiplication of filters that the operation • is commutative and associative. To prove that its neutral element is [e˙ lst], observe that the filter Fe˙ contains F for an arbitrary ⊂- least C-filter F. Therefore, Fe˙ lst ≈ Fe˙ ≈ F by Corollary 2.6 and by Proposition 2.2 from [1]. Hence, (Fe˙ lst)lst = F. (ii) By the same proposition, the inclusions λa(F) ⊂ a˙ F and a˙ x˙ ⊂ (ax˙ ) imply λa(F) ≈ a˙ F and a˙ x˙ ≈ (ax˙ ) for any a, x ∈ X and for any C-filter F on X. Hence, ˜λa(˜x) = i(a) • ˜x for any a ∈ X, ˜x ∈ X˜, and i(ax) = i(a) • i(x) for any a, x ∈ X. (iii) Let ˜x = [F], ˜y = [G] be arbitrary elements of X˜. Then ˜f (˜x • ˜y) corresponds to the ⊂-least C-filter on Y which is equivalent to f((FG)lst) ≈ f(FG) = f(F)f (G) by Proposition 2.1. By Corollary 2.6, this filter is equivalent ˜ ˜ to (f (F)lstf(G)lst)lst = f(˜x) • f (˜y). The fact that the function f → ˜f preserves identities and compositions of maps, was proved above. The proof is complete. D) Here, we begin to study properties of maps of X into topological spaces taking any C-filter on X into a convergent one. In particular, we prove that any such a map can be uniquely continuous extended to the finest unitary extension X˜. For Cauchy maps of T2-Cauchy spaces, it was showed in [3]. From now on, we will only consider two-sided C-filters again. Corresponding statements for left and right ones are similar. Definition 2.11. Let X be a Hausdorff topological monoid with an underlying set X, Y a topological space and f : X → Y a map. This map is said to be unitarily quasi-extending if lim f (F) consists of an only point for each C-filter F on X and each point of Y is the limit of such a filter. This definition implies that f(X) is dense in Y. By Proposition 1.4, Corollary 1.5 and Theorem 1.9, the map i : X → X˜ is unitarily quasi-extending. Proposition 2.12. Let f : X → Y be a unitarily quasi-extending map. Then: |X| (i) |Y| ≤ 22 .

(ii) Y is a T1-space. (iii) The map f is continuous in the unitary topology on X.

(iv) lim f(F1) = lim f(F2) for any equivalent C-filters F1, F2. (v) For any C-filter F on X and for any point y ∈ Y, if the filter f(F) clusters to y, then lim f(F) = y. Proof. (i) The cardinality of Y can not be more than the cardinality of the set of filters on X.

(ii) For y1, y2 ∈ Y with y1 ≠ y2 and an arbitrary filter F on X, if any neighborhood of y1 contains y2, then y2 ∈ lim F implies y1 ∈ lim F.

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(iv) By Proposition 2.3 from [1], the filter F1 ∩ F2 is a C-filter. Therefore, the filter f (F1 ∩ F2) converges. It implies that f(F1) and f(F2) have the same limit. (v) If f(F) clusters to y, then there exists a finer than F filter F′ on X such that f (F′) converges to y. By Proposition 2.2 from [1], F′ is a C-filter which is equivalent to F. Hence, F converges to y by statement (iv). (iii) Let F be a filter on X converging in the unitary topology to a point x. By Theorem 3.2 from [1], it is a C-filter which is equivalent to x˙. The singleton {f(x)} is a member of the filter f(x˙). Therefore, this filter converges to f(x). Then lim f (F) = f(x) again by (iv).

We shall now prove the universal property of the finest unitary extension. For the category of T2-Cauchy spaces, a similar statement was proved in [2]. Theorem 2.13. Let X = (X, m, τ) be a Hausdorff topological monoid, X˜ its finest unitary extension, i : X → X˜ the canonical injection and f : X → Y a unitarily quasi-extending map. Then there exists a unique continuous map ˜f : X˜ → Y such that f = ˜f ∘ i. This map is surjective. Proof. Let F be an arbitrary ⊂-least C-filter on X and ˜x = [F] the corresponding point of X˜. Set ˜f(˜x) = lim f (F). ˜ If, in particular, ˜x = [x˙ lst] = i(x), then f(i(x)) = lim f(x˙ lst) = f(x) follows, by statement (iii) of the preceding ˜ Proposition, from the fact that x˙ lst converges in the unitary topology to x. It means that f = f ∘ i. Show that ˜f is continuous in the natural topology. Let ˜x be an arbitrary point of X˜. First, assume that ˜x ∈ i(X). Then, by Theorem 1.9, i(X) is a neighborhood of ˜x, and ˜f coincides with f ∘ i−1 in this neighborhood. Hence, ˜f is continuous at ˜x by statement (ii) of Theorem 1.9 and by the preceding proposition. Let now ˜x = [F] ∈ X˜ \ i(X) and W an arbitrary neighborhood of y = ˜f(˜x) = lim f (F). By the preceding proposition, U = f −1(W) is open in the unitary topology and belongs to F since W ∈ f (F). Lemma 1.10 shows now that V = i(U) ∪ {[F]} is a neighborhood of ˜x. Moreover, ˜f (V) = f(U) ∪ {y} ⊂ W, i.e. ˜f is continuous at ˜x. The uniqueness of ˜f follows from the fact that i(X) is dense in X˜. ˜f is surjective since each point of Y is the limit of a filter of the form f(F) where F is a C-filter on X.

References

[1] B. G. Averbukh, On unitary Cauchy filters on topological monoids, Topol. Algebra Appl., 1 (2013), 46-59. [2] R. Fric, D. C. Kent, Completion functors for Cauchy spaces, Int. J. Math. & Math. Sci. 2, No. 4 (1979), 589-604. MR 80#54042. Zbl 428.54018. [3] H.H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334-341. MR 37#874. Zbl 155.50302 [4] D.C. Kent, G.D. Richardson, Regular completions of Cauchy spaces, Pacific Journal of mathematics 51, No. 2 (1974), 483- 490. [5] D. C. Kent, G. D. Richardson, Cauchy completion categories, Canad. Math. Bull. 32, No. 1 (1989), 78-84. MR 90#54007. Zbl 675.54004. [6] E. Lowen-Colebunders. Function Classes of Cauchy Continuous Maps, Pure and Applied Mathematics, Marcel Dekker Inc., New York, (1989). [7] J. F. Ramaley, O. Wyler, Cauchy spaces, (1968), http://repository.cmu. edu/math/97. [8] N. Rath, Completion of a Cauchy space without the T2-restriction on the space, Internat. J. Math. & Math. Sci. 24, No. 3 (2000), 163-172. [9] N. Rath, Completions of Filter Semigroups, Acta Math. Hungar. 107 (1-2) (2005), 45-54. [10] E. E. Reed, Completions of Uniform Convergence Spaces, Math. Ann. 194 (1971), 83-108. MR 45#1109. Zbl 217.19603. [11] O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970), 1-12. MR 44#985. Zbl 207.52603.

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