Actions of Semitopological Groups

Canad. Math. Bull. Vol. 62 (2), 2019 pp. 441–450 http://dx.doi.org/Ë§. Ëþh/CMB-z§ËÇ-§Ë§-þ ©Canadian Mathematical Society z§ËÇ

Jan van Mill and Vesko M. Valov

Abstract. We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.

1 Introduction

All spaces under discussion are Tychonoò. Continuous actions of semitopological groups are considered in this paper. Recall that a group G with a topology on the set G that makes the multiplication G × G → G separately continuous is called semitopological. A semitopological group G is ω-narrow [ËË] if for every neighborhood U of the neutral element e in G there is a countable set A ⊂ G with UA = AU = G. If not stated otherwise, we consider le actions θ∶ G × X → X, where G is a semitopological group and X is a space. We denote θ(g, x) by gx for all g ∈ G and x ∈ X. If θ is continuous (resp., separately continuous), we say that the action is continuous (resp., separately continuous). For any such action, we consider the translations θ X X θx G X θ x gx θx g gx g∶ → and the maps ∶ → deûned by g( ) = and ( ) = . _ese two types of maps are continuous when θ is separately continuous. We also say that θ acts transitively on X if all θx , x ∈ X, are surjective maps.If the action θ can be extended to a continuous action θ̃∶ G × X̃ → X̃, where X̃ is a compactiûcation of X, then X̃ is called an equivariant compactiûcation of X, or simply a G-compactiûcation. _e paper is motivated by the celebrated theorem of Uspenskij [Ëh,Ë ] that a compactum X is a Dugundji space provided X admits a continuous transitive action of an ω-narrow topological group. _ere is a growing interest recently in study- ing paratopological, quasitopological, or semitopological groups versus topological groups; see [z,h,ËË]. In that spirit, we provide a generalization of Uspenskij’s theorem in two directions. We consider spaces that are not necessarily compact and actions not necessarily by topological groups.It is not clear to us whether our conditions are all essential, but we will show that some are; see Example z.@. A space is k-separable if it contains a dense σ-compact set.Our main result, obtained by following Uspenskij’s method of proof described in [h], is the following theorem.

Received by the editors February z6, z§ËÇ; revised March 6, z§ËÇ. Published electronically May ËÇ, z§ËÇ. Author J. v. M. is pleased to thank the Department of Mathematics at Nipissing University for gen- erous hospitality and support. Author V. M. V. was partially supported by NSERC Grant z@ËË -Ëh. AMS subject classiûcation: þ HËË, þ HËþ, zzAË§. Keywords:Dugundji space, group action, semitopological group, topological group.

Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 16:37:35, subject to the Cambridge Core terms of use. 442 J. van Mill and V. M. Valov _eorem Ë.Ë Let X admit a continuous transitive action of an ω-narrow semitopological group G such that X has a G-compactiûcation. _en (i) X is skeletally Dugundji provided it contains a dense Čech-complete k-separable subspace; (ii) X is openly Dugundji if X is Čech-complete and σ-compact.

Recall that a continuous map f ∶ X → Y is skeletal [] if Int f (U)= ~ ∅ for every open U ⊂ X, where f (U) denotes the closure of f (U) in Y. Skeletally Dugundji spaces were introduced in [6]. In a similar way we deûne openly Dugundji spaces: A space X is skeletally (resp. openly) Dugundji if there exists a well ordered inverse S X pβ α β τ system = { α , α , < < } with surjective skeletal (resp., open) bonding maps, where τ is a cardinal, satisfying the following conditions (we identify the cardinal τ with the initial ordinal of cardinality τ): X pα+Ë (i) § is a separable metrizable space and all maps α have metrizable kernels i.e., M X ( there exists a separable metrizable space α such that α+Ë is embedded X M pα+Ë π X in α × α and α coincides with the restriction S α+Ë of the projection π X M X ∶ α × α → α ); γ τ X (ii) for any limit ordinal < the space γ is a (dense) subset of

X pβ α β γ lim{ α , α , < < }; ←Ð X S p X X α p S X (iii) is embedded in lim such that α ( ) = α for each , where α ∶ lim → α ←Ð ←Ð is the α-th limit projection; (iv) for every bounded continuous real-valued function f on lim S there exists α ∈ A g X f g p ←Ð and a continuous function on α with = ○ α . _e inverse system S is called almost continuous if it satisûes conditions (ii), and X is said to be the almost limit of S if condition (iii) holds, which we denote by X = a−lim S. ←Ð We also say that S is factorizing if it satisûes condition (iv). Dugundji spaces were introduced by Pelczynski [Ë§] as the compacta X such that for every embedding of X in another compactum Y, there is a regular linear extension operator u∶ C(X) → C(Y) between the Banach spaces of all continuous functions on X and Y.It was established by Haydon [Ç] that a compactum X is Dugundji if and only if X is the limit space of a well ordered inverse system satisfying the above conditions pβ X X with all α being open. Equivalently, is a Dugundji space if and only if is a compact openly Dugundji space. _ere is a tight connection between skeletally Dugundji and Dugundji spaces: X is skeletally Dugundji if and only if every compactiûcation of X is co-absolute with a Dugundji space; see [6, _eorem h.h]. Since any Baire space admitting a continuous and transitive action of an ω-narrow topological group G has a G-compactiûcation (see [þ, Ëh]), we have the following corollary. Corollary Ë.z Let a space X admit a continuous transitive action of an ω-narrow topological group. _en X is skeletally Dugundji (resp., openly Dugundji) provided it contains a dense Čech-complete k-separable subset (resp., X is Čech-complete and σ-compact).

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Note that Corollary Ë.z implies Uspenskij’s theorem mentioned above. It appears that continuous actions of semitopological groups on compact spaces can be reduced to continuous actions of topological groups. _is simple observation combined with Uspenskij’s theorem implies the following theorem. _eorem Ë.h If an ω-narrow semitopological group G acts continuously and transitively on a pseudocompact space X and G is a k-space, then βX is a Dugundji space.

We also consider separately continuous actions of semitopological or topological groups. Every separately continuous le action θ∶ G × X → X generates a right action Θ∶ G × C(X) → C(X), deûned by Θ(g, f )(x) = f (gx); see Lemma z.z.It is easily seen that this action is separately continuous when C(X) carries the pointwise C X C X convergence topology ( ( ) with this topology is denoted by p( )). We say that θ s-action C X G is an , if each orbit of Θ is a separable subset of p( ). For example, if is separable and θ is separately continuous, then θ is an s-action. According to Lem- masz .z and z.h, this is also true if θ is continuous, X is compact, and G is an ω-narrow semitopological group. _e conclusion of _eorem Ë.Ë remains true if continuity of the action is weakened to separate continuity, but with the additional requirement that θ be an s-action. _eorem Ë. Let X admit a separately continuous transitive action θ of an ω-narrow semitopological group G such that θ is extendable to a separately continuous s-action of G over a compactiûcation of X. _en X is skeletally Dugundji provided it contains a dense Čech-complete k-separable subspace.

For pseudocompact spaces, we have the following analogue of Uspenskij’s result [Ëh, _eorem z]. _eorem Ë.þ If a pseudocompact space X admits a separately continuous transitive s-action of an ω-narrow semitopological group, then X is skeletally Dugundji.

Concerning separately continuous actions of topological groups, we have the following fact.

Proposition Ë.@ If a pseudocompact (resp., Čech-complete and σ-compact) space X admits a separately continuous transitive action of an ω-narrow topological group, then βX is Dugundji (resp., X is openly Dugundji).

_e paper is organizing as follows: the proofs of _eorems Ë.Ë and Ë.h are given in Section z;Section h contains the proofs of _eorems Ë. and Ë.þ, and PropositionË. @. 2 Continuous Actions

Our ûrst lemma is a version of [h, Proposition Ë§.h.Ë]. Recall that a map f ∶ X → Y is nearly open if f (U) ⊂ Int f (U) for every open U ⊂ X; see [h].

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Lemma z.Ë Let θ∶ G × X → X be a separately continuous transitive action on a Baire space X. (i) If G is an ω-narrow semitopological group, then all maps θx ∶ G → X, x ∈ X, are skeletal; (ii) If, in addition to (i), G is a topological group, then θx are nearly open maps.

Proof (i) We ûrst prove that θx (U)= ~ ∅ for every neighborhood U of the identity e in G. Suppose x ∈ X and U is a neighborhood of e. Since G is ω-narrow, there is a countable set A ⊂ G with AU = G.It follows from the transitivity of θ that the map θx is surjective.Hence, {θx (gU) = gUx ∶ g ∈ A} is a countable cover of X, and so is {gUx ∶ g ∈ A}. Because X is a Baire space, there is g ∈ A with gUx Ux θ X X Int =~ ∅. Consequently, Int =~ ∅ (recall that the translation g∶ → is a homeomorphism). If U ⊂ G is a non-empty open set, we choose g ∈ U and a neighborhood V of e with gV ⊂ U. _en Int Vx =~ ∅, so Int gVx =~ ∅. Finally, since gVx ⊂ Ux, we have Int Ux =~ ∅. _erefore, θx is skeletal. (ii) _is item follows from the observation that the proof of [h, Proposition Ë§.h.Ë] remains true for separately continuous actions of ω-narrow groups on Baire spaces.

∗ If a semitopological group G acts on a space X, we say that a function f ∈ C (X) is right-uniformly continuous if for every ε > §, there is neighborhood O of the neutral element e such that for all g ∈ O and x ∈ X we have S f (x) − f (gx)S < ε. We denote X C∗ X the set of all right-uniformly continuous functions on by r,G ( ).

Lemma z.z Let X be a space and let θ∶ G × X → X be a continuous action of a semi- ∗ ∗ topological group G on X. _en the right action Θ∶ G × C (X) → C (X), Θ(g, f ) = f θ , is well deûned and each translation C∗ X C∗ X is a linear isometry ○ g Θg∶ ( ) → ( ) on the Banach space C∗ X . Moreover, if X is compact, then C X C∗ X . ( ) ( ) = r,G ( )

∗ Proof For every g ∈ G and f ∈ C (X), we have Θ(g, f )(x) = f (gx), x ∈ X. Because θ X X g f f C∗ X g∶ → is surjective, YΘ( , )Y = Y Y.It is obvious that each map Θg∶ ( ) → C∗ X f f θ ( ), Θg( ) = ○ g, is linear. So all Θg are linear isometries. Moreover, for every g h G f C∗ X g h f g f θ f θ θ , ∈ and ∈ ( ), we have Θ( , Θ( , )) = Θ( , ○ h) = ○ h ○ g. Since θ θ θ g h f f θ hg f h ○ g = hg, Θ( , Θ( , )) = ○ hg = Θ( , ). So Θ is a right action. Suppose X is compact.Let f ∈ C(X) and ε > §. For every x ∈ X, choose a W x X f x′ f x ε x′ W θ neighborhood x of in such that S ( ) − ( )S < ~z for all ∈ x . Because O G V W is continuous, there exists a neighborhood x of e in and a neighborhood x ⊂ x x X θ O V W V i k of in such that ( x × x ) ⊂ x .Let { xi ∶ = Ë, z, ... , } be a ûnite subcover of V x X O k O g O x X the cover { x ∶ ∈ }, and let = ⋂i=Ë xi . _en, for every ∈ and ∈ , there j k x V gx W f gx f x ε f x f x is ≤ with ∈ x j and ∈ x j .Hence, S ( )− ( j)S < ~z and S ( )− ( j)S < ε~z. Consequently, S f (gx) − f (x)S < ε, which completes the proof of the claim.

_e proof of next lemma is a slight modiûcation of the proof of [h, Lemma Ë§.h.z]; it is included for the sake of completeness.

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Lemma z.h Let θ∶ G ×X → X be a continuous action of an ω-narrow semitopological ∗ ∗ group on a space X and Θ∶ G × C (X) → C (X) be the action from Lemma z.z. _en ∗ the orbit f G = {Θ(g, f ) ∶ g ∈ G} is a separable subset of the Banach space C (X) for each f C∗ X . ∈ r,G ( ) f C∗ X n U Proof We ûx ∈ r,G ( ). _en for every ∈ N, there is a neighborhood n of g f f n g U G ω n e with YΘ( , ) − Y < Ë~ for all ∈ n. Because is -narrow, for each there A G U A G A ∞ A exists a countable set n ⊂ such that n n = .Hence, = ⋃n=Ë n is also countable and it suõces to show that f A = {Θ(a, f ) ∶ a ∈ A} is dense in f G. To g G n g U a A g g a this end, let ∈ and for each choose n ∈ n and n ∈ n with = n n. _en g f f f Θ( , ) = Θgn an ( ) = Θan (Θgn ( )). So g f a f f f f f YΘ( , ) − Θ( n , )Y = YΘg( ) − Θan ( )Y = YΘan (Θgn ( )) − Θan ( )Y. g f a f f f Because Θan is an isometry, we obtain YΘ( , ) − Θ( n , )Y = YΘgn ( ) − Y = g f f n g f Af YΘ( n , ) − Y < Ë~ . _erefore, every neighborhood of Θ( , ) meets , which is as required.

For any compact space X let H(X) be the homeomorphism group of X with the compact-open topology.It is well known [Ë] that H(X) is a topological group. We need the following observation.

Lemma z. Let X be a compact space admitting a continuous action θ∶ G × X → X of a semitopological group G. _en the homomorphism φ∶ G → H(X), deûned by φ(g) = θ , is continuous. g

Proof Let K ⊂ X be compact and let U ⊂ X be open. Consider the subbasic open − set [K, U] = {h ∈ H(X) ∶ h(K) ⊂ U} in H(X). Take an arbitrary g ∈ φ Ë([K, U]). φ g θ K U gK U θ−Ë U G X _en ( ) = g ∈ [ , ], hence ⊂ . Consider the open set ( ) in × . It contains {g} × K. So by the compactness of K, there is a neighborhood V of g in − ′ ′ G such that V × K ⊂ θ Ë(U).Hence, g ∈ V implies g K ⊂ U, which means that − V ⊂ φ Ë([K, U]). Proof of _eorem Ë.Ë Suppose X contains a dense Čech-complete k-separable subspace D. Suppose also that θ∶ G × X → X is a continuous transitive action of a semitopological ω-narrow group on X such that θ can be extended to a continuous action θ̃∶ G × Y → Y, where Y is a compactiûcation of X. According to Lemma z. , the image φ(G) is a topological ω-narrow subgroup of H(Y). Since H(Y) acts continuously on Y, so does φ(G). On the other hand, φ(G) acts transitively on X, because G does. _erefore, we can assume that G is an ω-narrow topological group such that θ̃ is a continuous action on Y, and its restriction θ is a continuous and transitive action on X. According to Lemma z.z, each f ∈ C(Y) is right-uniformly continuous. As above, θ̃(g, y) is denoted by g y for all g ∈ G and y ∈ Y. _e action θ̃ generates a right continuous action Θ̃∶ G × C(Y) → C(Y), deûned by Θ̃(g, f )(y) = f (g y) (see Lemma z.z). D k σ Z Z ∞ F Because is -separable, it contains a dense -compact set .Let = ⋃i=Ë i and Y D ∞ Y F Y F Y i j ∖ = ⋃i=Ë i , where i and i are compact sets. Since i ∩ j = ∅, for each , there f C Y f F f Y C g f g G i j is i j ∈ ( ) with i j( i ) ∩ i j( j) = ∅.Let § = {Θ̃( , i j) ∶ ∈ , , = Ë, z, ...}

Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 16:37:35, subject to the Cambridge Core terms of use. 446 J. van Mill and V. M. Valov C Y C f γ τ τ C Y C and ( ) ∖ § = { γ ∶ < }, where is the cardinality of ( ) ∖ §. For every car- α τ C C g f g G γ α C Y C dinal § < < , deûne α = § ∪ {Θ̃( , γ) ∶ ∈ , < }. _en ( ) = ⋃§≤α<τ α Y C(Y) y Y f y and is embedded in R by identifying each ∈ with ( ( )) f ∈C(Y). We con- p Y Cα α pβ Y Y β α sider the natural projections α ∶ → R for ≥ § and α ∶ β → α for > , Y p Y Y p Y p pβ where β = β( ) and α = α ( ).Obviously, all α and α are continuous and p pβ p β α p y f y y Y α τ α = α ○ β for > .Observe that α ( ) = ( ( )) f ∈Cα for each ∈ and < . Because every f ∈ C(Y) is right-uniformly continuous, we can apply Lemma z.ht o conclude that the orbits Γ( f ) = {Θ̃(g, f ) ∶ g ∈ G}, f ∈ C(Y), are separable subsets C Y C C Y p Y Y of ( ).Hence, § is also a separable subset of ( ) and §( ) = § is a metric X p X Z p Z α Z X compactum. Let α = α ( ) and α = α ( ) for all ≥ §. Because is dense in , Z X every α is dense in α . Claim Ë X Every α contains a dense Čech-complete subspace, and hence it is a Baire space.

L Y ∞ p Y L p−Ë L L Indeed, let § = § ∖ ⋃i=Ë §( i ) and = § ( §). _en § is Čech-complete, Z L X Z L X and we have the inclusions § ⊂ § ⊂ § and ⊂ ⊂ . Moreover, the map p L L L L C C p−Ë p L L §S ∶ → § is perfect, so is also Čech-complete. Since § ⊂ α , α ( α ( )) = p L L L p L L and the map α S ∶ → α = α ( ) is a perfect surjection. So α is a dense Čech- X complete subspace of α . Claim z p Y Y Each α ∶ → α is a skeletal map. C i.e., G C C θ G Y Y Since α is Θ̃-invariant ( Θ̃( , α ) = α ), there is an action ̃α ∶ × α → α , θ̃ g′ p y f g′ y deûned by α ( , α ( )) = ( ( )) f ∈Cα , which makes the diagram below commutative.

̃θ G × Y ÐÐÐÐ→ Y × × × id ×p × pα × α × Ö Ö ̃ G Y θ α Y × α ÐÐÐÐ→ α

θ θ G X X p X θ θ Moreover, the restriction α = ̃α S × α is an action on α such that ( α S )○ = α . θ θ Let show that α is separately continuous. For this, we only need that ̃ is separately g G θ g continuous. Indeed, ûx ∈ . Clearly, (̃α ) is continuous by commutativity of the y Y z Y diagram and compactness of all spaces involved. Now, ûx α ∈ α . Pick ∈ such p z y θ yα p θz that α ( ) = α . _en ̃α is equal to α ○ ̃ and hence is continuous, being the composition of two continuous functions. θ X θ X Note that, since is transitive on , each α acts transitively on α . To show that p U Y X Y x U X α is skeletal, let ⊂ be open. Since is dense in , we can ûx ∈ ∩ . _en the θx G X θ y G X p θx θ y y p x maps ∶ → and α ∶ → α are continuous and α ○ = α , where = α ( ).

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So we have the commutative diagram

x G θ / X

pα y θ α X α . X θ y Since α is Baire space, according to Lemma z.Ë(ii), α is nearly open, and θ y θx −Ë U X p U X p U X p U X α (( ) ( ∩ )) = α ( ∩ ) implies α ( ∩ ) ⊂ IntXα clXα α ( ∩ ).Hence, p X X X X Y U X every α S ∶ → α is nearly open. Because α is dense in α and ∩ is dense in U p U p , IntYα clYα α ( )= ~ ∅. _us, α is a skeletal map.

Claim h pα+Ë Each map α has a metrizable kernel.

f f q Y Γ( fα ) Let Γ( α ) be the orbit of α and let α ∶ → R be the projection. Since, by f C Y Y q Y Lemma z.h, Γ( α ) is a separable subspace of ( ) and is compact, the image α ( ) p p q pα+Ë is a metric compactum. Finally, because α+Ë is the diagonal product α △ α , α S Y pβ α β τ has a metrizable kernel. Note that the inverse system Y = { α , α , < < } is well Y p Y S ordered such that § is a metric compactum, all maps α are skeletal, and = lim Y . S C C ←Ðα τ _e system Y is continuous, because α = ⋃γ<α γ for every limit cardinal < . S Y Moreover, it follows from our construction that Y is factorizable. So is a skeletally Dugundji space. Finally, since X is dense in Y, the space X is skeletally Dugundji as well (see [6, _eorem h.h]). To prove _eorem Ë.Ë(ii), observe that if X is Čech-complete and σ-compact, then L X α p X X X α = α for all and the maps α S ∶ → α are perfect and nearly open (see the proof of Claim z). Because every closed nearly open map is open, we ûnally obtain p X S X tβ α β τ that all α S are open. _en the inverse system X = { α , α , < < } is contin- tβ tβ pβ X tα+Ë uous such that α are open and perfect maps, where α = α S β. Moreover α have X X metrizable kernels and § is a Polish space. _erefore, is openly Dugundji. We will now show that _eorem Ë.Ë does not hold for spaces on which we do not P-space G impose extra conditions. A is a space in which every δ -subset is open. Lemma z.þ Every skeletally Dugundji P-space is discrete.

Proof Let X be a skeletally Dugundji P-space, and assume that x is a non-isolated point of X. Consider βX.It consequently contains the non-isolated P-point x. As- sume that βX is co-absolute with a Dugundji space Z. _en there is a compact space Y which admits irreducible maps f ∶ Y → βX and g∶ Y → Z (irreducible maps are − surjective by deûnition). _e set A = f Ë({x}) is a nowhere dense closed P-set in Y since f is irreducible. We claim that Y does not satisfy the countable chain condition. α ω U Y Indeed, by recursion on < Ë, we will construct a nonempty open subset α of U A U U β α ω such that α ∩( ∪⋃β<α β) = ∅. Suppose that we deûned β for every < < Ë. V Y U A A _en = ∖ ⋃β<α β is a neighborhood of .Hence, since is nowhere dense, U Y U V A U there is a nonempty open subset α of such that α ⊆ ∖ . _en, clearly, α is U α ω as required.It now suõces to observe that the collection { α ∶ < Ë} witnesses the fact that Y does not satisfy the countable chain condition. But this means that Z does

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not satisfy the countable chain condition since g is irreducible. _is is a contradiction, since Z is Dugundji and hence dyadic [h, Ë§.Ë.h].

Example z.@ Let X be the one-point Lindelöõcation of a discrete space of size ω X P F X Ë. _en is a -space, and hence so is its free topological group ( ) [h, 6. .6]. Moreover, ûnite products of X are Lindelöf, hence F(X) is Lindelöf [h, 6.Ë.ËÇ]. _is means that F(X) is a Lindelöf P-group and hence, in particular, is ω-narrow. Since X is not discrete, F(X) has no isolated points. By Teleman [Ëz], F(X) has an F(X)-com- pactiûcation. But F(X) is not skeletally Dugundji by the lemma just proved.

_e following question remains open.

Question z.6 Let the (k-separable) Baire space X admit a continuous transitive action of an ω-narrow semitopological group G such that X has a G-compactiûcation. Is X skeletally Dugundji?

Proof of _eoremË. h Suppose X is a pseudocompact space and θ∶ G × X → X is a continuous transitive action, where G is an ω-narrow semitopological group that is a k-space. _en there is a continuous action θ̃∶ G × βX → βX extending θ. Indeed, θ X X θ since each g∶ → is a homeomorphism, g can be extended to a homeomorphism θ βX βX θ G βX βX θ g x θ x ̃g∶ → .Hence, we can deûne ̃∶ × → by ̃( , ) = ̃g( ).Obviously, θ g θ g x θ g g x θ ̃( Ë, ̃( z, )) = ̃( Ë z, ). So it remains to show that ̃ is continuous. To this end, observe that G × βX is a k-space as a product of the k-space G and the compactum βX (see [@, _eorem h.h.z6]). _erefore, it is enough to show that each restriction θ θ K βX K G ̃K = ̃S( × ) is continuous, where ⊂ is compact. And this is true, because by [@, _eorem h.Ë§.z@] the product K × X is pseudocompact.Hence, by Glicksberg’s theorem [@, h.Ëz.z§(c)], β(K × X) = K × βX. _us, θS(K × X) has a continuous K βX θ extension to × and it is obvious that this extension coincides with ̃K . Now we can complete the proof of _eoremË. h. We apply Lemma z. to the action θ̃ and obtain that φ∶ G → H(βX) is a continuous homomorphism. So φ(G) is a an ω-narrow topological group acting continuously on βX. Denote this action by θ φ G βX βX θ φ G X ̃φ∶ ( ) × → .Obviously ̃φ, restricted on ( ) × , provides a continuous θ φ G X X X X and transitive action φ∶ ( ) × → on . _erefore, is a pseudocompact space admitting a continuous and transitive action of an ω-narrow topological group. Hence, we can apply [Ëh, _eorem z] to conclude that βX is a Dugundji space. 3 Separately Continuous Actions

First we prove the following lemma.

Lemma h.Ë Let F C X be a separable subset and let p X F be the diagonal ⊂ p( ) ∶ → R product of all h ∈ F. _en p(X) is a sub-metrizable space.

Proof Take a countable dense subset Γ of F and let φ∶ X → RΓ be the diagonal product of all h ∈ Γ. _en φ(X) is a metrizable space. _ere is a continuous surjection λ p X φ X p x h x p X φ x ∶ ( ) → ( ) assigning to each point ( ) = ( ( )h∈F ) ∈ ( ) the point ( ) =

Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 16:37:35, subject to the Cambridge Core terms of use. Actions of Semitopological Groups 449 h x φ X λ λ p x λ p x ( ( )h∈Γ) ∈ ( ). We claim that is one-to-one. Suppose ( ( Ë)) = ( ( z)) for p x p x p X h F h x h x the distinct points ( Ë), ( z) ∈ ( ). So there is ∈ with ( Ë)= ~ ( z), and h x h x η F let S ( Ë) − ( z)S = . Since Γ is dense in with respect to the pointwise topology, f f x h x η i f x f x there exists ∈ Γ such that S ( i ) − ( i )S < ~z for = Ë, z.Hence, ( Ë)= ~ ( z), φ x φ x which contradicts the equality ( Ë) = ( z).

Proof of _eorem Ë. We follow the proof of _eorem Ë.Ë.Let X contain a dense Čech-complete k-separable space D and let θ∶ G × X → X be a transitive separately continuous action of an ω-narrow semitopological group such that θ can be extended to a separately continuous s-action θ̃∶ G × Y → Y, where Y is a compactiûcation of X. Suppose also that Z is a dense σ-compact set of D. _e action θ̃ generates G C Y C Y a right separately continuous action Θ̃∶ × p( ) → p( ) such that each orbit f g f g G f C Y C Y Γ( ) = {Θ̃( , ) ∶ ∈ }, ∈ ( ), is a separable subset of p( ). As in the Y C(Y) C C proof of _eorem Ë.Ë, we embed in R , deûne the sets § and α and consider p Y Cα α X the projections α ∶ → R , ≥ §. According to ClaimË , each α is a Baire space. θ θ θ G X X Because ̃is separately continuous and is transitive, the actions α ∶ × α → α are p also separately continuous and transitive; see Claim z. _e proof that α are skeletal maps is the same. _e only diòerence is, according to Lemma z.Ë(i), that the map θ y G X y X U Y x U X α ∶ → α is skeletal for every ∈ α . So for every open ⊂ and points ∈ ∩ y p x θy θx −Ë and = α ( ), we have IntXα clXα α (( ) (U∩X)) = IntXα clXα pα (U∩X)= ~ ∅. _is p implies that α is skeletal; see the proof of Claim z. Finally, in the proof of Claim h we pα+Ë can use Lemma h.Ë to conclude that every α has a metrizable kernel. Proof of _eoremË. þ Suppose X is a pseudocompact space, G is an ω-narrow semitopological group, and θ∶ G × X → X is a separately continuous transitive s-action. θ G C X C X _en generates a separately continuous action Θ∶ × p( ) → p( ) such that f g f g G f C Y C Y each orbit Γ( ) = {Θ( , ) ∶ ∈ }, ∈ ( ), is a separable subset of p( ). We follow the proof of _eorem Ë.Ë, considering X instead of Y. We embed X in C(X) C X f γ τ C C g f g G γ α R , where ( ) = { γ ∶ < }, and let α = § ∪ {Θ( , γ) ∶ ∈ , < } C X with § being the set of all constant functions on . We also consider the projections p X X Cα θ G X X α τ α ∶ → α ⊂ R , and the transitive actions α ∶ × α → α , § ≤ < .Observe X f C θ that § is a point, because all ∈ § are constant functions. Since is separately con- θ X tinuous, so is each α (see the proof of Claim z). Because every α is a Baire space (being pseudocompact), following the proof of Claim z and using Lemma z.Ë(i), we p show that all maps α are skeletal. pα+Ë f It remains to show that the maps α have metrizable kernels.Let Γ( α ) = g f g G f q X Γ( fα ) {Θ( , α ) ∶ ∈ } be the orbit of α and let α ∶ → R be the projection. Since f C X q X Γ( α ) is a separable subset of p( ), by Lemma h.Ë, α ( ) is sub-metrizable. On q X q X the other hand, α ( ) is pseudocompact. So by [Ëþ], α ( ) is a metric compactum. S X pβ α β τ _erefore, the inverse system = { α , α , < < } is well ordered, almost continu- X pα+Ë ous, consists of skeletal maps with § being a point and each α having a metrizable kernel. Moreover, S is factorizable and X = a−lim S.Hence, X is a skeletally Dugundji ←Ð space.

Proof of PropositionË. @ Suppose θ∶ G × X → X is a separately continuous transitive action on a Baire space X, where G is an ω-narrow topological group. _en, by

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Lemma z.Ë(ii), each θx ∶ G → X is nearly open. Consequently, by [ , Proposition z], θ is continuous.If X is Čech-complete and σ-compact,Corollary Ë.z implies that X is openly Dugundji. In the case where X is pseudocompact, we apply [Ëh, _eorem z] to conclude that βX is Dugundji. References

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