Admissible topologies for groups of homeomorphisms and substitutions of groups of G-spaces

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Admissible topologies for groups of homeomorphisms and substitutions of groups of G-spaces ✩ A. Karassev a , K.L. Kozlov b,∗,1 a

Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada b Department of General Topology and Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia

a r t i c l e

i n f o

Article history: Received 31 January 2019 Received in revised form 8 June 2019 Accepted 9 June 2019 Available online xxxx MSC: 54H15 57S05 22F30 54D35

a b s t r a c t We present approaches to substitute the acting groups of G-spaces that preserve various properties of actions, such as transitivity, being a coset space, and preserving a fixed equiuniformity in case of a G-Tychonoff space. Such substitutions make it possible to obtain bounds on cardinal invariants of acting groups with admissible topologies, and to investigate metrizability of phase spaces and find their equivariant completions of the same weight. © 2020 Elsevier B.V. All rights reserved.

Keywords: Homogeneity Topological group G-space Uniformity Completion

1. Introduction A topological space X is homogeneous if for any x, y ∈ X there exists a homeomorphism h of X such that h(x) = y. The group of homeomorphisms in discrete topology acts continuously and transitively on a homogeneous space X. Unfortunately, the discrete topology of an acting group limits us in the usage of topological methods to investigate the groups of homeomorphisms, their actions, and the properties of space X. Admissible topologies (in the sense of R. Arens) on (sub)groups of homeomorphisms expand our opportunities. R. Arens [3] called a topology for the group G of homeomorphisms of X admissible if G is a ✩

The work is supported by NSERC Grant 257231-09.

* Corresponding author. 1

E-mail addresses: [email protected] (A. Karassev), [email protected] (K.L. Kozlov). The second author is partially supported by RFBR Grant 15-01-05369.

https://doi.org/10.1016/j.topol.2019.107033 0166-8641/© 2020 Elsevier B.V. All rights reserved.

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topological group in this topology and the action α : G × X → X, α(g, x) = g(x), is continuous. The survey of A. V. Arhangel’skii and J. van Mill [5] can serve as a good introduction to the subject of the paper. Let a topological group G acts continuously on X. We look for admissible topologies on G in order to do the following: • obtain restrictions on cardinal invariants of topological groups (that realize homogeneity of a space (i.e. acts transitively)) (Corollaries 3.3, 3.5, 3.10, 3.18, and Examples 3.19, 3.24, 3.28, 3.31); • find possible values of cardinal invariants of admissible topologies on acting groups (that realize homogeneity of a space) (Theorems 3.12 and 3.13, Corollaries 3.14, 3.17 and 3.20 of §3.3 and §3.4); • give conditions on (infra)metrizability of spaces X (Theorem 3.27, Corollaries 3.23, 3.30); • find equiuniformity of the same weight as the character of space X and equivariant completion of X preserving weight (Corollaries 3.16, 3.20, 3.26). Looking for admissible topologies on G we use the following two approaches. I. We look for the class of topological groups G for which G is isomorphically embedded in the product of groups from G. Using the product structure we find the subproduct into which G is isomorphically and continuously mapped and use the induced topology on G. II. We look for uniformity on X which allows us to examine the topology of uniform convergence on G. The main general results are Theorems 3.2 and 3.8 of §3.1 and §3.2. In Theorem 3.2 we combine both approaches. Corollary 3.4 is a special case of τ -narrow and τ -balanced acting groups (range-G classes of groups with respect to the classes G of groups of weight and character ≤ τ respectively). In section 2, we provide information about G-spaces and uniformities on G-spaces. In §2.4 we present properties of equivariant pairs of maps. In §2.5 we describe properties of G-range class of groups, and in §2.6 we introduce cardinal invariants which characterize isomorphic embeddings and isomorphic mappings of G-range groups into products of groups from G. We examine groups with restrictions on weight or character. In the countable case we examine separable metrizable, metrizable and Čech complete groups. In Theorem 3.1 the acting group is substituted by the quotient group with respect to the kernel of action. This allows us to omit the trivial part of the action. 2. Preliminaries All spaces are assumed to be Tychonoff, maps are continuous and notations, terminology and designations are from [8]. Nbd(s) stands for open neighborhood(s); clX A and int X A are the closure and interior of a subset A of a space X, respectively. By d(X), χ(X), ψ(X), w(X) we denote the density, character, pseudocharacter, and weight of a space X, respectively. A metrizable space is called Polish if it is separable and has a complete metric. The identity map is denoted by id. For information about topological groups see [4] and [21]. By NG (e) we denote the family of nbds of the unit e ∈ G. We use notation H < G for a closed and H
G for an invariant closed subgroup H of G. The kernel of a continuous homomorphism ϕ : G → H is denoted by ker ϕ, ker ϕ
G. For H
G, G/H is the quotient group. If K is a compactum then Hom(K) is the group of homeomorphisms of K in compact open topology. 2.1. G-spaces By an action of a group G on a set X we mean a map α : G × X → X such that

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α(e, x) = x, α(g, α(h, x)) = α(gh, x) for any x ∈ X, g, h ∈ G (where e is the unit of the group G). We denote by αx : G → X, x ∈ X, and αg : X → X, g ∈ G, the maps defined by the rules αx (g) = α(g, x) = αg (x). If it is clear what action is considered, we write α(g, x) := gx. The action is transitive if the map αx is a surjection and the set of orbits X/G is a one point set. For a set X with an action α of a group G the subgroup H = {g ∈ G : gx = x, x ∈ X} is called the kernel of action and is denoted kerα . If kerα = {e}, then the action is called effective. For a family of subsets γ = {Oα : α ∈ A} of X and g ∈ G we set gγ = {gOα : α ∈ A}. A family of covers Γ = {γs : s ∈ S} is saturated if gγs ∈ Γ for any s ∈ S, g ∈ G. A G-space is a triple (G, X, α), consisting of a space X with a fixed continuous action α : G × X → X of a topological group G. The kernel kerα of a continuous action is a closed invariant subgroup. Let (G, X, αG ) and (H, Y, αH ) be G-spaces. A pair of maps (ϕ : G → H, f : X → Y ) such that ϕ : G → H is a homomorphism and the diagram ϕ×f

G × X −→ ↓ αG X

f

−→

H ×Y ↓ αH Y

is commutative is called an equivariant pair of maps of G-spaces, notation (ϕ, f ) : (G, X, αG ) → (H, Y, αH ). The commutativity of the diagram is equivalent to the following condition f (gx) = ϕ(g)f (x) for any x ∈ X, g ∈ G. If f is an embedding then the equivariant pair (ϕ, f ) is an equivariant embedding of (GX , X, αX ) into (GY , Y, αY ). The composition of equivariant pairs of maps is an equivariant pair. A G-space (G, X, α) is called G-Tychonoff if there is an equivariant embedding (id, f ) of (G, X, α) into a G-space (G, bX, α ˜ ) where bX is a compactification of X. The triple (G, bX, α ˜ ), together with the embedding (id, f ), is a G-compactification of (G, X, α). The maximal G-compactification is denoted (G, βG X, αβ ). 2.2. d-Open actions Information about (d-)open actions, their properties and natural uniform structures which are generated by (d-)open actions can be found in [13], [14] and [15]. For the convenience of the reader we remind some facts which are used below. Definition 2.1. ([13]) An action α : G × X → X is called open if x ∈ int (Ox) for any point x ∈ X and O ∈ NG (e); d-open if x ∈ int (cl(Ox)) for any point x ∈ X and O ∈ NG (e). Remark 2.2. Terminology in Definition 2.1 is motivated by the fact that an action is “open” (“d-open”) iff maps αx : G → X, x ∈ X, are open (d-open) [15, Remark 1.7]. F. Ancel [1] called continuous open actions micro-transitive and continuous d-open actions weakly microtransitive. If (G, X, α) is with a d-open action then X is a direct sum of clopen subsets (components of the action). Each component of the action is the closure of the orbit of an arbitrary point of this component. If an action is open then X is a direct sum of clopen subsets which are the orbits of the action [15, Remark 1.7]. For a G-space (G, X, α) with an open action and one component of action X is the quotient space of G.

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The following theorem is a generalization of a correspondent result of F. Ancel for complete metrizable groups [1]. Theorem 2.3. [14, Theorem 3] Let (G, X, α) be a G-space with a d-open action and let for any x ∈ X the coset space G/Gx be Čech complete. Then the action is open. In particular, a d-open continuous action of a Čech complete group is open. 2.3. Uniform structures on G-spaces Uniform structures on spaces are introduced by the families of covers [11] and are compatible with their topology. The weight w(U) of a uniformity U is the minimum cardinality of its base. For covers u and v of X we write u  v (u∗  v) if u refines (star-refines) v. A uniformity U on X for a G-space (G, X, α) is called equiuniformity (see, for example, [18]) if it is saturated (i.e. that αg is uniformly continuous, g ∈ G) and bounded (i.e. that for any u ∈ U there exist O ∈ NG (e) and v ∈ U such that Ov = {OV : V ∈ v}  u). A G-space (G, X, α) is G-Tychonoff iff there is an equiuniformity on X. The action on a G-Tychonoff space (G, X, α) with equiuniformity U can be extended ˜ of X with respect to to the continuous action of the Raikov completion ρG of G on the completion X the equiuniformity U [18, Theorem 3.1]. By equivariant completion of (G, X, α) we understand the G-space ˜ α ˜ α (G, X, ˜ ) with the natural equivariant embedding (id, i) : (G, X, α) → (G, X, ˜ ). Theorem 2.4. [12] If the action on a G-space (G, X, α) is d-open, then the family of covers γO = {int (cl(Ox)) : x ∈ X}, O ∈ NG (e), is the base of the maximal equiuniformity on X. 2.4. Properties of equivariant pair of maps Proposition 2.5. Let (ϕ, id) : (G, X, αG ) → (H, X, αH ) be an equivariant pair of maps of G-spaces and ϕ : G → H be a continuous epimorphism. Then: (a) (b) (c) (d) (e)

ker ϕ ⊂ kerαG ; ϕ(kerαG ) = kerαH and if αG is effective then αH is effective; Gx = Hx for any x ∈ X, in particular, if αG is transitive then αH is transitive; if αG is open (d-open) then αH is open (d-open) and components of actions αG and αH coincide; if U is an equiuniformity on X in (H, X, αH ) then U is an equiuniformity on X in (G, X, αG ) and, hence, if (H, X, αH ) is G-Tychonoff then (G, X, αG ) is G-Tychonoff.

Proof. Properties (a), (b) and (c) immediately follow from the equality αG (g, x) = αH (ϕ(g), x), x ∈ X, g ∈ G. In order to prove property (d) take x ∈ X and O ∈ NH (e). For U = ϕ−1 (O) ∈ NG (e) x ∈ int (U x) = int (ϕ(U )x) = int (Ox) in case of an open action αG , x ∈ int (cl(U x)) = int (cl(ϕ(U )x)) = int (cl(Ox)) in case of a d-open action αG . Thus openness (d-openness) is preserved. Since int (cl(Gx)) = int (cl(Hx)), x ∈ X, the components of actions αG and αH coincide. Now we prove property (e). If U is an equiuniformity on X in (H, X, αH ) then it is a uniformity on X. ϕ(g) g From the equality αG = αH for any g ∈ G it follows that U is a saturated uniformity in (G, X, αG ). Its boundedness follows from the equality αG (ϕ−1 (O), x) = αH (O, x), x ∈ X, O ∈ NH (e). The last statement in property (e) is a straightforward consequence. 2

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Remark 2.6. If G is a discrete group then a G-space (G, X, α) is G-Tychonoff. This yields the existence of an example of a G-Tychonoff space whose image under an equivariant pair of maps is not G-Tychonoff. From Proposition 2.5 (e) we obtain the following corollary. Corollary 2.7. Let (G, X, α) be a G-Tychonoff space and U be the maximal equiuniformity on X. If (ϕ, id) : (G, X, αG ) → (H, X, αH ) is an equivariant pair of maps and U is an equiuniformity on X in (H, X, αH ), then U is the maximal equiuniformity on X in (H, X, αH ). 2.5. Range-G groups Let G be a class of topological groups. A class G of topological groups is called τ -multiplicative (τ is an infinite cardinal) if the product of τ representatives of G is an element of G. If τ = ℵ0 then we say the class is σ-multiplicative. If the class is closed under arbitrary products then it is called multiplicative and if under arbitrary finite products then it is called finitely-multiplicative. A class G of topological groups is called hereditary if a subgroup of a representative of G is an element of G. A class G of topological groups is closed under quotients if G/H ∈ G for any H
G. A topological group G is called range-G if for any O ∈ NG (e) there exists a continuous homomorphism h of G to H ∈ G such that h−1 (U ) ⊂ O for some U ∈ NH (e) [4, §3.4]. Remark 2.8. Let G be a class of topological groups. The following conditions are equivalent for a group G: • G is range-G; • G has a family of homomorphisms to topological groups from G which separates points and closed sets. A range-G topological group is • topologically isomorphic to a subgroup of a Tychonoff product of a family of topological groups from G [4, Theorem 3.4.21]. If, additionally, G is a finitely-multiplicative class then all three above conditions are equivalent. Proposition 2.9. Let G be a class of topological groups. Then the class of range-G groups is hereditary. If G is finitely-multiplicative then the class of range-G groups is multiplicative. If G is hereditary and closed under quotients then the class of range-G groups is hereditary and closed under quotients. Proof. Only condition of being closed under quotients is not trivial. Let G ∈ range-G, H
G, ϕ : G → G/H be the quotient map. The idea of the proof is the following. If {hα : G → Tα : Tα ∈ G, α ∈ A} is the family of homomorphisms that separates points and closed sets for G, then {fα : G/H → Tα /cl(hα (H)) : α ∈ A} is the family of homomorphisms that separates points and closed sets for G/H. Take O ∈ NG/H (e). There exist T ∈ G, a continuous epimorphism h : G → T and U ∈ NT (e) such that −1 h (U ) ⊂ ϕ−1 (O). Put K = ker h, K
G.

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Evidently, h−1 (U )H ⊂ ϕ−1 (O). From the equality h−1 (U )HK = h−1 (U )H we have h−1 (U )H = h−1 (h(h−1 (U )H)) = h−1 (U h(H))) and h−1 (U h(H)) ⊂ ϕ−1 (O).  Since cl(h(H)) = {W h(H) : W ∈ NT (e)}, we have h−1 (cl(h(H))) ⊂ ϕ−1 (O). For W ∈ NT (e) such that W 2 ⊂ U we have h−1 (W cl(h(H))) ⊂ h−1 (W 2 h(H)) ⊂ ϕ−1 (O). Since h(HK) ⊂ cl(h(H)) and homomorphism h is continuous, we have h(cl(HK)) ⊂ cl(h(H)) and cl(HK) ⊂ h−1 (cl(h(H))). Obviously, cl(HK)
G and h−1 (cl(h(H)))
G. Therefore, (G/cl(HK))/(h−1 (cl(h(H)))/cl(HK)) and (G/K)/(h−1 (cl(h(H)))/K) are topologically isomorphic to G/h−1 (cl(h(H))) and the quotient map G/cl(HK)→(G/K)/(h−1 (cl(h(H))/ K)) is well defined. For the one-to-one continuous homomorphism ν : G/K → T we have ν(h−1 (cl(h(H)))/K) = cl(h(H)). Hence, the continuous homomorphism G/cl(HK) → T /cl(h(H)) as a composition of the quotient map G/cl(HK) → (G/K)/(h−1 (cl(h(H)))/K) and the map of the quotient group (G/K)/(h−1 (cl(h(H)))/K) onto T /cl(h(H)) is well defined. Since H
cl(HK), the continuous homomorphism f : G/H → T /cl(h(H)) as composition of homomorphisms G/H → G/cl(HK) → T /cl(h(H)) is well defined. Let ψ : T → T /cl(h(H)) be the quotient map. Evidently, ψ(W ) = V ∈ NT /cl(h(H)) (e), ψ −1 (V ) = W cl(h(H)) and T /cl(h(H)) ∈ G. Then h−1 (ψ −1 (V )) = h−1 (W cl(h(H))) ⊂ ϕ−1 (O). Thus ϕ(h−1 (ψ −1 (V ))) ⊂ ϕ(ϕ−1 (O)) = O. For arbitrary O ∈ NG/H (e) there exist T /cl(h(H)) ∈ G, V ∈ NT /cl(h(H)) (e) and a continuous homomorphism f : G/H → T /cl(h(H)) such that f −1 (V ) = ϕ(h−1 (ψ −1 (V ))) ⊂ O (since f ◦ ϕ = ψ ◦ h). Hence, G/H is a range-G group. 2 By ib(G) and inv(G) we denote, respectively, the index of narrowness and the invariance number of a topological group G. Note that inv(G) ≤ ib(G), see [4, Chapter 5]. If ϕ : G → H is a continuous epimorphism then ib(H) ≤ ib(G).

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The following τ -multiplicative, hereditary and closed under quotients classes G of topological groups and corresponding multiplicative, hereditary and closed under quotients classes of range-G topological groups are well-known [4, §5.1]. Class G of groups

Range-G class of groups

groups of weight ≤ τ

τ -narrow groups (or groups G with ib(G) ≤ τ )

groups of character ≤ τ

τ -balanced groups (or groups G with inv(G) ≤ τ )

As could be observed from the table every τ -narrow group is τ -balanced. An inframetrizable group is a subgroup of a Čech complete group [21]. The following σ-multiplicative, hereditary and closed under quotients classes G of topological groups and corresponding multiplicative, hereditary and closed under quotients classes of range-G topological groups are well-known [4, §3.4], [21, Chapter 13]. Class G of groups

Range-G class of groups

groups of countable weight (or separable metrizable groups)

ω-narrow groups

groups of countable character (or metrizable groups)

ω-balanced groups

inframetrizable groups (or subgroups of Čech complete groups)

subgroups of products of Čech complete groups

As could be observed from the table every ω-narrow group is ω-balanced. Every ω-balanced group is a subgroup of a product of Čech complete groups. 2.6. Indexes of topological and continuous isomorphisms of a range-G group Definition 2.10. For a class G of topological groups and G ∈ range-G we define the index of embedding by itG (G) = inf{|F| : F is a family of homomorphisms to topological groups from G which separates points and closed sets}, and the index of injection by icG (G) = inf{|F| : F is a family of homomorphisms to topological groups from G which separates points} Example 2.11. A σ-product of a countable product of separable metrizable groups [4, §1.6, Ch. 1] in boxtopology is an example of an ω-narrow non metrizable group G. Hence, itG (G) = icG (G) for the class G of ω-narrow groups. Proposition 2.12. Let G be a class of topological groups and G ∈ range-G. The following hold: • icG (G) ≤ itG (G); • itG (H) ≤ itG (G) (icG (H) ≤ icG (G)) for any subgroup H of G; • itG (Πs∈S Gs ) ≤ sup{|S|, itG (Gs ) : s ∈ S} for a finitely-multiplicative class G and Gs ∈ range-G (icG (Πs∈S Gs ) ≤ sup{|S|, icG (Gs ) : s ∈ S} for Gs ∈ range-G); • itG (G/H) ≤ itG (G) for a hereditary and closed under quotients class G, H
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Proof. The first two statements are evident. In order to prove the third statement it is sufficient to consider homomorphisms which are compositions of projections on finite subproducts Πnk=1 Gsk and the product of homomorphisms defined on Gsk , k = 1, . . . , n (homomorphisms which are compositions of projections on factors Gs and homomorphisms defined on Gs ). The proof of the fourth statement follows from the proof of Proposition 2.9. 2 Question 2.13. Let G be a hereditary and closed under quotients class of topological groups, G ∈ range-G and H
G. Is icG (G/H) ≤ icG (G)? Proposition 2.14. Let G be a class of groups of character (weight) ≤ τ and G be a τ -balanced (τ -narrow) topological group. Then • icG (G) · τ = ψ(G); • itG (G) · τ = χ(G). Proof. Since inv(G) ≤ τ , by [4, §5.1, Lemma 5.1.6] there is a continuous isomorphism of G into Πs∈S Gs , where Gs is a group of character (weight) ≤ τ , |S| ≤ τ iff ψ(G) ≤ τ . Since inv(G) ≤ τ , by [4, §5.1, Lemma 5.1.6] there is an isomorphic embedding of G into Πs∈S Gs , where Gs is a group of character (weight) ≤ τ , |S| ≤ τ iff χ(G) ≤ τ [4, §5.1, Lemma 5.1.6]. 2 Corollary 2.15. Let G be a class of groups of character (weight) ≤ τ and G be a τ -balanced (τ -narrow) topological group. Then • ψ(G) ≤ τ iff G is continuously isomorphic to a group from G; • χ(G) ≤ τ iff G ∈ G. Note that an arbitrary topological group is a quotient group of a topological group of countable pseudocharacter [20]. Question 2.16. Let G be τ -balanced (τ -narrow) topological group, H
G. Is ψ(G/H) ≤ ψ(G)? Question 2.17. Let G be a class of inframetrizable groups, G ∈ range-G, and τ be an uncountable cardinal. When itG (G) ≤ τ (icG (G) ≤ τ )? 3. Substitutions of groups of G-spaces Theorem 3.1. Assume that (G, X, αG ) is a G-space, H = kerαG . Then (G/H, X, αG/H ) is a G-space with effective action, where αG (g, x) = αG/H (ϕ(g), x), x ∈ X, g ∈ G, ϕ : G → G/H is the quotient map. Additionally, if (G, X, αG ) is G-Tychonoff and U is an equiuniformity (respectively the maximal equiuniformity) on X, then (G/H, X, αG/H ) is G-Tychonoff and U is an equiuniformity (respectively the maximal equiuniformity) on X. Proof. The first statement is easy to check. For the second statement, if (G, X, αG ) is G-Tychonoff and U is an equiuniformity on X, then for any u ∈ U there exists O ∈ NG (e) such that {αG (O, x) : x ∈ X}  u.

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Then ϕ(O) ∈ NG/H (e) and {αG/H (ϕ(O), x) : x ∈ X} = {αG (O, x) : x ∈ X}  u. Thus, U is an equiuniformity on X of (G/H, X, αG/H ). Corollary 2.7 can be used to show the maximality of U in case U is the maximal equiuniformity on X of (G, X, αG ). 2 3.1. Admissible topologies for range-G homeomorphisms groups Theorem 3.2. Let (G, X, αG ) be a G-space with effective action, G be a finitely-multiplicative class of topological groups, and G be a range-G group with itG (G) ≥ κ. If a) |X/G| ≤ κ and χ(X) ≤ κ or b) (G, X, αG ) is G-Tychonoff and U is an equiuniformity (respectively the maximal equiuniformity) on X with w(U) ≤ κ, then icG (G) ≤ κ and there exists an admissible group topology T on G (weaker than or equal to the original one) such that: • H = (G, T ) is a range-G group, itG (H) ≤ κ (in particular H ∈ G if G is a τ -multiplicative class of topological groups, τ ≥ κ); • in case b) (H, X, αH ) is G-Tychonoff and U is an equiuniformity (respectively the maximal equiuniformity) on X. Proof. Since G is a range-G group we may assume that G is a subgroup of a Tychonoff product Π = Π{Gk ∈ G : k ∈ K} and fK = Δ{fk : k ∈ K} is an embedding, where {fk : k ∈ K} is a separating family of homomorphisms to topological groups from G. In case a) fix a point xl , l ∈ L, |L| ≤ κ, from each orbit and take a base of nbds {Utl : t ∈ Tl }, |Tl | ≤ κ, of xl , l ∈ L. For each Utl take Otl ∈ NG (e) such that Otl Url ⊂ Utl for some nbd Url of xl . In case b) fix a base {Ut : t ∈ T }, |T | ≤ κ, of equiuniformity U. For each Ut take Ot ∈ NG (e) such that {Ot x : x ∈ X}  Ut . In both cases, without loss of generality, we may assume that each Otl or Ot is a trace on G of a rectangular set in Π which depends on the finite number of coordinates Ktl , t ∈ Tl , l ∈ L, or Kt , t ∈ T . The cardinality   of the set S = {Ktl : t ∈ Tl , l ∈ L} or S = {Kt : t ∈ T } is κ. Let ΠS = Π{Gs ∈ G : s ∈ S}, prS : Π → ΠS , H = prS (G) ⊂ ΠS . Evidently, H is range-G (H ∈ G if G is a τ -multiplicative class of topological groups, τ ≥ κ}), ϕ = prS |G is a continuous homomorphism of G onto H, and ϕ(Otl ) or ϕ(Ot ) is open nbd of unit in H, since ϕ−1 (ϕ(Otl )) = Otl for all t ∈ Tl , l ∈ L, or ϕ−1 (ϕ(Ot )) = Ot for all t ∈ T . Moreover, ϕ ◦ fK = Δ{fs : s ∈ S}. Further, ϕ is a monomorphism. Indeed, for any g ∈ G, g = e, there is x ∈ X such that αG (g, x) = gx = x. In case a) there is xl , l ∈ L, such that x = αG (h, xl ) = hxl and αG (g, hxl ) = g(hxl ) = hxl . Hence, αG (h−1 gh, xl ) = (h−1 gh)xl = xl . There exists Otl ∈ NG (e) such that (h−1 gh)xl ∈ / Otl xl . Since ϕ is a −1 l l −1 l homomorphism and ϕ (ϕ(Ot )) = Ot , ϕ(h )ϕ(g)ϕ(h) ∈ / ϕ(Ot ) and ϕ(g) ∈ / ϕ(h)ϕ(Otl )ϕ(h−1 ) ∈ NH (e). Therefore, g ∈ / ker ϕ. In case b) there is Ut ∈ U such that gx ∈ / St(x, Ut ). Hence, for Ot ∈ NG (e) we have gx ∈ / Ot x and ϕ(g) ∈ / ϕ(Ot ) ∈ NH (e). Therefore, g ∈ / ker ϕ. In both cases ker ϕ = {e}.

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For x ∈ X, g ∈ H put αH (g, x) = αG (g, x). Since ϕ ◦ fK = Δ{fs : s ∈ S}, the monomorphism ϕ may be considered to be the identity map of G onto H. Therefore, the action αH of H on X is well-defined. Further, (H, X, αH ) is a G-space in case a). Indeed, every αH h : X → X, h ∈ H, is continuous. It remains to prove continuity of action αH at points {e} × X ⊂ H × X. For x = αH (g, xl ) = gxl and its nbd W the set g −1 W is a nbd of xl . There is a nbd Url of xl and Otl ∈ NG (e) such that Otl Url ⊂ g −1 W . The set gUrl is a nbd of gxl = x. There is V ∈ NH (e) such that g −1 V g ⊂ Otl . Then for V ∈ NH (e) and nbd gUrl of x we have V gUrl ⊂ gOtl Url ⊂ gg −1 W = W . In case b) (H, X, αH ) is a G-Tychonoff space and U is an equiuniformity on X in (H, X, αH ). In fact, every αH h : X → X, h ∈ H, is uniformly continuous with respect to U and for every u, v ∈ U, v  u, there is Ot ∈ NG (e) such that {Ot x : x ∈ X}  v (this yields that the action αH is bounded by U) and Ot v  u. Hence, Ot St(x, v) ⊂ St(x, u) and Ot Int(St(x, v)) ⊂ Int(St(x, u)) for any x ∈ X. This yields that the action αH is continuous. The maximality of U in the correspondent suppositions follows from Corollary 2.7. 2 Corollary 3.3. There is no continuous and effective action on X of a range-G topological group G with icG (G) > κ such that either |X/G| ≤ κ and χ(X) ≤ κ or (G, X, α) is G-Tychonoff and there exists an equiuniformity U on X with w(U) ≤ κ. From property (c) of Proposition 2.5, Theorem 3.2, Corollary 2.7 and facts from §2.5 we obtain the following statement.

Corollary 3.4. Assume that (G, X, αG ) is a G-space with effective action, G is a τ -narrow group (respectively a τ -balanced group), χ(G) ≥ κ. If a) |X/G| ≤ κ and χ(X) ≤ κ or b) (G, X, αG ) is G-Tychonoff and U is an equiuniformity (respectively the maximal equiuniformity) on X with w(U) ≤ κ, then ψ(G) ≤ κ and there exists an admissible group topology T on G (weaker than or equal to the original one) such that: • H = (G, T ) is a τ -narrow group, w(H) ≤ max{τ, κ} (respectively a τ -balanced group, χ(H) ≤ max{τ, κ}); • and in case b) (H, X, αH ) is G-Tychonoff and U is an equiuniformity (respectively the maximal equiuniformity) on X. Moreover, in case a) for an action of a τ -narrow group we have d(X) ≤ τ · |X/G| · χ(X). Corollary 3.5. Let χ(X) ≤ τ . There is no continuous and effective action on X of a topological group G with ψ(G) > τ , inv(G) ≤ τ , for which |X/G| ≤ τ . Proof. Let G acts continuously and effectively on X, inv(G) ≤ τ and |X/G| ≤ τ . Then by Theorem 3.12 there exists an admissible group topology T on G such that H = (G, T ) acts continuously, effectively on X and χ(H) ≤ τ . Then, ψ(G) ≤ τ . 2

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3.2. Admissible topologies as topologies of uniform convergence with respect to equiuniformities Theorem 3.6. [10, Theorem 2.2] If X possesses a uniform structure U with respect to which every element of some homeomorphism group G is uniformly continuous, then G is a topological group relative to the uniform convergence induced by the uniformity U. Remark 3.7. The sets Ou = {g ∈ G : ∀x ∈ X g(x) ∈ St(x, u)}, u ∈ U, form a base of nbds (not open) at e. Hence, χ(G) ≤ w(U). The uniformity of uniform convergence in Theorem 3.6 coincides with the right uniformity on G [21, Chapter 2, Exercise 2]. Uniformity U in Theorem 3.6 is an equiuniformity and, hence, (G, X, α) is a G-Tychonoff space [2, Remark 4.2 (c)]. The following theorem is a generalization of [2, theorem 4.6]. Theorem 3.8. [16, Lemma 6] Let (G, X, α) be a G-Tychonoff space with an effective action, and U be an equiuniformity on X. Then (1) the topology of uniform convergence τU on G is coarser than the original one, χ(H = (G, τU )) ≤ w(U), (2) (H, X, α) is a G-Tychonoff space and U is an equiuniformity on X. R. Arens showed that the compact open topology on the homeomorphisms group of a compactum is the coarsest admissible one [3]. See also paper of Yu.M. Smirnov [22] where this result is sharpened. The next result has the same motivation. Corollary 3.9. Let U be a uniformity on X. For any admissible group topology on G for which (G, X, α) is a G-Tychonoff space with equiuniformity U on X, the topology of uniform convergence τU on G is the coarsest one on G. Corollary 3.10. Let U be a uniformity on X, w(U) ≤ τ . There is no continuous and effective action on X of a topological group G with ψ(G) > τ for which U is an equiuniformity on X. Proof. Let G acts continuously and effectively on X, U be an equiuniformity on X such that w(U) ≤ τ . Then χ(H = (G, τU )) ≤ τ , (H, X, αH ) is a G-Tychonoff space and U is an equiuniformity on X for the topology of uniform convergence τU . Hence, ψ(G) ≤ τ . 2 A bijection f of uniform spaces is called a uniform equivalence if f and f −1 are uniformly continuous. Proposition 3.11. If U is a totally bounded uniformity on X, then w(G) ≤ w(U) for the group of uniform equivalences G of a uniform space (X, U) endowed with the topology τU of uniform convergence. Proof. The completion bX of X with respect to U is a compactum. Since each homeomorphism g ∈ G is uniformly continuous, it admits a unique extension g  : bX → bX which is uniformly continuous with respect to the unique uniformity on bX. Clearly, (g ◦ h) = g  ◦ h and idX = idbX . In other words, the map g → g  is a monomorphism of the group G into the group Hom(bX). Thus, identifying G with its image in Hom(bX), we can assume that G is a subgroup of Hom(bX). Then the topology of the uniform convergence on G coincides with the compact open topology [2, Proposition 4.4]. The latter topology is of weight ≤ w(bX). The rest follows from the coincidence of weight w(bX) with the weight of the unique uniformity of bX which restriction on X is U. Hence, w(bX) = w(U). 2

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3.3. Substitutions of groups of G-spaces (general case) Theorem 3.12. For a G-space (G, X, αG ) there exist a G-space (H, X, αH ) such that χ(H) ≤ χ(X) · inv(G) · |X/G|, ib(H) ≤ ib(G), w(H) ≤ χ(X) · ib(G) · |X/G|, and an equivariant pair of maps (ϕ, id) : (G, X, αG ) → (H, X, αH ), where ϕ is a continuous epimorphism such that ker ϕ = ker αG . If the action αG is effective, then there exists an admissible group topology T on G (weaker or equal than the original one) such that H = (G, T ). Proof. Character, weight and invariance number are preserved when passing to a quotient group. Thus, an equivariant pair of maps (ϕ, id) : (G, X, αG ) → (G/ kerαG , X, αG/ kerαG ) makes it possible to assume, without loss of generality, that the action αG is effective. If χ(G) ≤ χ(X) · |X/G|, then H = G. Otherwise, the group G is a subgroup of the product of a family of topological groups of character ≤ inv(G) [4, Theorem 5.1.9]. By Corollary 3.4 there exists an admissible group topology T on G (weaker than or equal to the original one) such that (H = (G, T ), X, αH ) is a G-space and χ(H) ≤ χ(X) · inv(G) · |X/G|, ib(H) ≤ ib(G). Hence, w(H) = χ(H) · ib(H) · |X/G| ≤ χ(X) · inv(G) · ib(G) · |X/G| = χ(X) · ib(G) · |X/G|. Evidently, (H, X, αH ) is an equivariant image of (G, X, αG ).

2

The presence of a uniform structure on a space allows to use topology of uniform convergence on a subgroup of homeomorphisms as in §3.2. Theorem 3.13. For a G-Tychonoff space (G, X, αG ) with equiuniformity (respectively the maximal equiuniformity) U there exist a G-Tychonoff space (H, X, αH ) with equiuniformity (respectively the maximal equiuniformity) U such that χ(H) ≤ w(U), ib(H) ≤ ib(G), w(H) ≤ w(U) · ib(G), and an equivariant pair of maps (ϕ, id) : (G, X, αG ) → (H, X, αH ), where ϕ is a continuous epimorphism such that ker ϕ = ker αG . If the action αG is effective, then there exists an admissible group topology T on G (weaker than or equal to the original one) such that H = (G, T ). Proof. The same reasons as in the proof of Theorem 3.12 makes it possible to assume, without loss of generality, that the action αG is effective. From Theorem 3.8 it follows that the topology of uniform convergence τU on G is weaker than the original topology. Further, H = (G, τU ) is a topological group by Theorem 3.6, (H, X, αH ) (αH (g, x) = αG (g, x) for g ∈ G, x ∈ X) is a G-Tychonoff space, and U is an equiuniformity on X by Remark 3.7. The maximality of U follows from Corollary 2.7. Thus, ib(H) ≤ ib(G).

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Since the sets Ou = {g ∈ G : ∀x ∈ X g(x) ∈ St(x, u)}, u ∈ U, are a base of nbds (not open) at e, χ(H) ≤ w(U). The last inequality follows from [4, Theorem 5.2.3]. 2 The following corollary is a consequence of Theorem 3.13 and Proposition 3.11. Corollary 3.14. Assume that for a G-Tychonoff space (G, X, αG ) U is a totally bounded equiuniformity on X. Then there exist a G-Tychonoff space (H, X, αH ) with equiuniformity U such that

w(H) ≤ w(U), and an equivariant pair of maps (ϕ, id) : (G, X, αG ) → (H, X, αH ), where ϕ is a continuous epimorphism such that ker ϕ = ker αG . If the action αG is effective, then there exists an admissible group topology T on G (weaker that or equal to the original one) such that H = (G, T ). Remark 3.15. Note that if the action αG : G × X → X in Theorems 3.12 and 3.13 is transitive or (d-)open, then the action αH : H × X → X is transitive or (d-)open by Proposition 2.5. Corollary 3.16. Let χ(X) ≤ τ , and X be a coset space of a group G or there is a d-open action of G on X such that |X/G| ≤ τ . (a) If G is a τ -balanced group, then there exists an equiuniformity U on X as on a coset space of a group of character ≤ τ such that w(U) ≤ τ (or there is a d-open action on X of a group of character ≤ τ ). ˜ α ˜ G ) of (G, X, αG ) such If w(X) ≤ τ , then, additionally, there exists an equivariant extension (G, X, ˜ ˜ that w(X) ≤ τ , where X is the completion of X with respect to U. (b) If G is a τ -narrow group, then, with X being a coset space of a group of weight ≤ τ , w(X) ≤ τ (or there is a d-open action on X of a group of weight ≤ τ with at most τ components of action). Proof. The same reasons as in the proof of Theorem 3.12, coupled with item (d) of Proposition 2.5 and closeness under quotients of the examined classes of groups, makes it possible to assume, without loss of generality, that the action αG is effective. Proof of the first statement in (a). By Corollary 3.4 there exists an admissible group topology T on G (weaker than or equal to the original one) such that χ(H = (G, T )) ≤ τ and its action is d-open. By [13, Proposition 4] the weight of the maximal equiuniformity U on X in (H, X, αH ) is ≤ τ . Evidently, U is an equiuniformity on X in (G, X, αG ). Proof of the second statement in (a). Since χ(X) ≤ w(X) ≤ τ , it follows from the above that there exists an equiuniformity U on X in (G, X, αG ) such that w(U) ≤ τ . The completion of X with respect to U is ˜ of X of weight τ and by [18, Theorem 3.1] there exists a natural the required equivariant completion X ˜ ˜ extension α ˜ G : G × X → X of action αG . Proof of (b). The same reasons as above makes it possible to assume, without loss of generality, that the action αG is effective, there exists an admissible group topology T on G (weaker than or equal to the original one) such that χ(H = (G, T )) ≤ w(H) ≤ τ , and its action is d-open. The weight of the maximal equiuniformity U on X in (H, X, αH ) is ≤ τ and there is a base of U of cardinality ≤ τ of open covers of cardinality ≤ τ . Hence, w(X) ≤ τ . 2

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3.4. Substitutions of groups of G-spaces (countable case) The following corollary is a straightforward consequence of Corollary 3.4 for actions of ω-narrow and ω-balanced groups, Theorem 3.2 and facts from §2.5 for actions of groups which are subgroups of products of Čech complete groups. Corollary 3.17. Assume that (G, X, αG ) is a G-space, G is an ω-narrow group (respectively ω-balanced group or a subgroup of a product of Čech complete groups). If |X/G| ≤ ℵ0 and χ(X) ≤ ℵ0 , then there exist: a G-space (H, X, αH ) such that H is a separable metrizable (respectively metrizable or inframetrizable) group; and an equivariant pair of maps (ϕ, id) : (G, X, αG ) → (H, X, αH ), where ϕ is a continuous epimorphism such that ker ϕ = ker αG . If, additionally, the action αG is effective, then there exists an admissible group topology T on G (weaker than or equal to the original one) such that H = (G, T ). In the case of an action of an ω-narrow group on a space X of countable character for which |X/G| ≤ ℵ0 (in particular, transitive), X is a separable space. Corollary 3.18. Let χ(X) ≤ ℵ0 . There is no continuous, effective and transitive action on X of an ω-balanced group G with uncountable pseudocharacter. There is no continuous and transitive action on X of an ω-narrow group if X is not separable. Example 3.19. An example of a homogeneous space X which is not a coset space is constructed in [10, §5 Two examples]. It is easy to show that χ(X) = ℵ0 , and the cellularity of X is uncountable. Hence, X is not separable. Therefore, there is no continuous and transitive action on X of an ω-narrow group. Applying theorem of extension of action to the completion of a phase space with respect to an equiuniformity [18, Theorem 3.1], we obtain the following corollary as a straightforward consequence of Theorem 3.13 and facts from §3.2. Corollary 3.20. Assume that (G, X, αG ) is a G-Tychonoff space, X is metrizable and U is an equiuniformity on X, w(U) ≤ ℵ0 . Then there exist: a G-Tychonoff space (H, X, αH ) with effective action such that H is metrizable (separable metrizable if G is ω-narrow), U is an equiuniformity on X; an equivariant pair of maps (ϕ, id) : (G, X, αG ) → (H, X, αH ), where ϕ is a continuous epimorphism such that ker ϕ = ker αG ; ˜ = X if U is com˜ α ˜ is metrizable (X and an equivariant completion (H, X, ˜ H ) of (H, X, αH ) where X plete). If, additionally, the action αG is effective, then there exists an admissible group topology T on G (weaker than or equal to the original one) such that H = (G, T ).

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The following corollary follows from Corollary 3.20, theorem of extension of action to the completion of a phase space with respect to an equiuniformity and Raikov completion of a group [18, Theorem 3.1], complete metrizability of a Raikov complete metrizable group, [14, Theorem 3] and [2, Theorem 5.10]. Corollary 3.21. Assume that (G, X, αG ) is a G-Tychonoff space with a d-open action and one component of action, X is metrizable and U is a complete equiuniformity on X, w(U) ≤ ℵ0 . Then X is a coset space of a complete metrizable group. If, additionally, X is a separable metrizable space, then X is a coset space of a Polish group. Example 3.22. In [19] J. van Mill constructed a homogeneous Polish space Z which is not a coset space and proved that no ω-narrow group acts transitively on Z by a separately continuous action [19, Corollary 3]. We note the following. A) There is no complete metric on Z with respect to which the group G of uniform equivalences, endowed with the topology of uniform convergence, acts d-openly and has one component of action. Indeed, if it were the case, there would exist a complete equiuniformity U with w(U) ≤ ℵ0 and by Corollary 3.21 Z would be a coset space. This is a contradiction. B) There is no totally bounded metric on Z with respect to which the group G of uniform equivalences, endowed with the topology of uniform convergence, acts transitively. Indeed, otherwise G would be separable metrizable [2], which is a contradiction with the properties of homogeneity of Z. From Corollary 3.17, condition (d) of Proposition 2.5 and [13, Corollary 4] (in which, among other things, metrizability of a phase space X of a G-space (G, X, α) is established, where G is metrizable and α is a d-open action) we obtain the following statement. Corollary 3.23. Let χ(X) ≤ ℵ0 , X be a coset space of a group G or there is a d-open action of G on X such that |X/G| ≤ ℵ0 . (a) If G is a subgroup of a product of Čech complete groups, then X with the maximal equiuniformity is an inframetrizable space [15, Remark 2.6] as a coset space of an inframetrizable group (or since X admits a d-open action of an inframetrizable group). (b) If G is an ω-balanced group, then X is metrizable as a coset space of a metrizable group (or since X admits a d-open action of a metrizable group). (c) If G is an ω-narrow group, then X is separable metrizable as a coset space of a separable metrizable group (or since X admits a d-open action of a separable metrizable group with no more than countable components of action). Example 3.24. Since the Sorgenfrey line S is not metrizable, there is no d-open action on S of an ω-balanced group G. Hence, the Sorgenfrey line is not a coset space of an ω-balanced group. The Sorgenfrey line is not Čech complete. Since a d-open action of a Čech complete group is open [14, Theorem 3] and a coset space of a Čech complete group is Čech complete [6, Theorem 2], there is no d-open action of a Čech complete group on the Sorgenfrey line. Question 3.25. Can the Sorgenfrey line be a coset space of a subgroup of a product of Čech complete groups (or, equivalently, of an inframetrizable group)? Applying theorem of extension of action to the completion of a phase space with respect to an equiuniformity [18, Theorem 3.1], and in the case of actions of ω-narrow groups compactification theorem [17, Theorem 2.13] we have.

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Corollary 3.26. Let χ(X) ≤ ℵ0 , X be a coset space of a group G or there is a d-open action of G on X such that |X/G| ≤ ℵ0 . If G is an ω-balanced (respectively ω-narrow) group, then there exists an equiuniformity (respectively totally bounded equiuniformity) U on X such that w(U) ≤ ℵ0 . ˜ α ˜ for Moreover, there exists an equivariant extension (G, X, ˜ ) of (G, X, α) and a complete metric on X which the induced uniformity is an extension of U (respectively G-compactification (G, bX, α ˜ ) of (G, X, α) where bX is a metrizable compactum). Theorem 3.27. Let X be a compact coset space of a subgroup of a product of Čech complete groups (or there is a continuous d-open action of a subgroup of a product of Čech complete groups with one component of action and |X/G| ≤ ℵ0 ), χ(X) ≤ ℵ0 . Then X is metrizable. Proof. By item (a) of Corollary 3.23 X is a coset space of an inframetrizable group G (or there is a continuous d-open action with one component of action of an inframetrizable group on X and |X/G| ≤ ℵ0 ). Then (G, X, α) is a G-Tychonoff space with a natural open action α in the first case, and d-open action in the second case. By [18, Theorem 3.1] the extension α ˜ ρ : ρG × X → X of action α is well defined and d-open. Here ρG, the Raikov completion of G, is a Čech complete group and by [14, Theorem 3] the action α ˜ ρ is open. Hence, X is a coset space of a Čech complete group. By [7, Propositions 3.2.1, 3.2.2] X is also a coset space of an ω-narrow group. Item (c) of Corollary 3.23 finishes the proof. 2 Example 3.28. The two-arrows space [8, Exercises 3.10.C] is not a coset space of a subgroup of a product of Čech complete groups. Remark 3.29. From the proof of Theorem 3.27 we can deduce a positive answer to Question 5.7 from [15] in case of compacta X with χ(X) ≤ ℵ0 . Corollary 3.30. Let X be a homogeneous space with Baire property, χ(X) ≤ ℵ0 . If there is a continuous transitive action of an ω-narrow group on X, then X is a separable metrizable space. Proof. In [23] it is proved that a transitive action α of an ω-narrow group G on a space X with Baire property is d-open. Hence, (G, X, α) is a G-Tychonoff space with a transitive d-open action. By item (c) of Corollary 3.23 X admits a transitive d-open action of a separable metrizable group. In [13, Corollary 4] it is shown that a d-open action of a metrizable group yields that the space X is metrizable. Moreover, if the acting group G is separable and the action is transitive then the phase space X is separable. 2 Example 3.31. There is no continuous and transitive action of an ω-narrow group on the two-arrows space and the homogeneous compactum X of V.V. Fedorchuk [9] (which is not a coset space and is of countable character). The result about the two-arrow space can also be deduced from [23], since the two-arrows space is not dyadic. The Sorgenfrey line has the Baire property. There is no continuous and transitive action of an ω-narrow group on the Sorgenfrey line. Acknowledgements The authors would like to express their gratitude to reviewer for valuable suggestions which improved exposition. References [1] F.D. Ancel, An alternative proof and applications of a theorem of E.G. Effros, Mich. Math. J. 34 (1) (1987) 39–55.

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