The SIN property in homeomorphism groups

Topology and its Applications 251 (2019) 94–106

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Topology and its Applications www.elsevier.com/locate/topol

The SIN property in homeomorphism groups Keith Whittington Department of Mathematics, University of the Pacific, Stockton, CA 95211, United States of America

a r t i c l e

i n f o

Article history: Received 6 April 2018 Received in revised form 29 October 2018 Accepted 30 October 2018 Available online 3 November 2018 MSC: 54C05 54E15 54H11

a b s t r a c t Characterizations are given for a group of homeomorphisms of a uniform space, with weakly micro-transitive action, to be SIN. If the phase space is compact and the group is given the compact-open topology, the group is SIN if and only if it is precompact. Under very general conditions, the group is shown to be SIN if and only if it is uniformly equicontinuous with respect to a particular uniformity on the phase space. The SIN property of any Hausdorff topological group is characterized by its actions. A new condition is discovered for the group topology to be admissible. © 2018 Elsevier B.V. All rights reserved.

Keywords: Topological group SIN Balanced Uniformity Equicontinuous Almost open Metric Micro-transitive d-open

0. Introduction The topology of an acting group and the properties of its action are useful tools in the investigation of G-spaces, and in particular, to the study of topological homogeneity. For instance, see the survey on this topic in [3]. This paper began as a naive investigation into how well one can coordinate a metric on a group G of homeomorphisms of a compact metric space X with a metric on X. Here we are giving h(X), the group of all self homeomorphisms of X, the compact-open topology. If D is a metric on X, the usual sup-metric d(a, b) = sup{D(ax, bx) | x ∈ X} E-mail address: kwhittington@pacific.edu. https://doi.org/10.1016/j.topol.2018.10.010 0166-8641/© 2018 Elsevier B.V. All rights reserved.

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gives a metric on h(X) compatible with this topology. Notice that d is right-invariant; i.e., d(ac, bc) = d(a, b). Furthermore, for all (a, b) ∈ G × G and all x ∈ X, we have D(ax, bx) ≤ d(a, b) However, if G is transitive, the existence of metrics d and D such that d is left-invariant and has this property places severe restrictions on both G and X. For instance, it is equivalent to G, the closure of G in h(X), being compact, and to X being isometrically homogeneous; namely, each element of G becoming an isometry under some metric for X. We also find that it is equivalent to G being a SIN-group, so this gives rise to a class of groups where being SIN is equivalent to being precompact; i.e., subgroups of h(X) that act transitively on X. This intriguing observation led to an investigation between the various properties in a more general setting using uniform spaces. Surprisingly, the study gives rise to a new condition for a topology on a homeomorphism group to be admissible. The only use we made of G being transitive was that this makes each map Tx open, where Tx : G → X is given by Tx (a) = ax [16, Theorem 2.1]. We were subsequently able to reduce this requirement to each Tx being almost open. The following theorem, whose proof will be deferred to the end, states the motivating result. A metric on G is invariant if it is both right and left invariant. When we refer to a “metric on a space”, we always mean a metric compatible with its topology. It should be mentioned that the equivalence of (4) and (5) is a theorem of Eilenberg from 1937, [17], and places no requirement on the maps Tx. Similarly, the equivalence of (3) and (4) is a classical result. Theorem. If X is a compact metric space, h(X) is given the compact-open topology, G is a subgroup of h(X), and each Tx is almost open, then the following statements are equivalent. (1) G and X have metrics d and D such that d is left-invariant and D(ax, bx) ≤ d(a, b) for all a, b ∈ G and all x ∈ X. (2) G and X have metrics d and D such that d is left-invariant and for every  > 0 there exists some δ > 0 such that for all a, b ∈ G with d(a, b) < δ and all x ∈ X, D(ax, bx) < . (3) G is equicontinuous. (4) X has a metric making each element of G an isometry. (5) G is compact. (The closure of G in h(X).) (6) G is precompact. (7) G is a SIN group. (8) G and X have metrics d and D satisfying D(ax, bx) ≤ d(a, b) and such that d is invariant. In [7, Theorem 4], Kozlov and Chatyrko give concise descriptions of a Tychonoff G-space X in the two scenarios that each Tx is open, or that each Tx is almost open. If each Tx is open, the orbits are closed-open sets, each of which is homeomorphic to a coset space of G. If each Tx is almost open, then X is disjoint union of closed-open sets, each of which is a union of orbits, and each orbit is dense in the closed-open set to which it belongs. 1. Preliminaries All topological spaces, including topological groups, are assumed to be Hausdorff. The symbols X, E, G, and θ will be reserved as follows, though their specificity will vary. Throughout, (X, E) will be a uniform space; that is, X is a topological space and E is a diagonal uniformity on X compatible with its topology. This means that the sets

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E[x] = {y ∈ X | (x, y) ∈ E}, where E ∈ E, form a neighborhood base at x ∈ X. It necessarily follows that X is a Tychonoff space, because being Tychonoff is equivalent to the existence of a compatible uniformity [10, Theorem 8.1.20]. G will always be a topological group with identity element e, and U(e) will denote the neighborhood system at e in G. We will also consider additional topologies on G under which it may not be a topological group. An element U ∈ U(e) is called invariant, if it is invariant under inner automorphisms; i.e., for each a ∈ G, aU a−1 = U . We call a subset, S ⊆ G, ω-narrow if for each U ∈ U(e) there is a countable set S  ⊆ S such that S ⊆ SU . We will always assume the existence of an action θ of G on X that is continuous in the second variable. To say that θ is an action, or that G acts on X (via θ), means that θ is a function from G × X to X which, with the convention of writing ax as short for θ(a, x), satisfies: ex = x and a(bx) = (ab)x for all a, b ∈ G and all x ∈ X. Each a ∈ G naturally induces a bijection θa : X → X by defining θa (x) = ax. To say that θ is continuous in the second variable means that each θa is continuous. It easily follows that each θa is a homeomorphism. The triple (G, X, θ) is transitive (or G acts transitively on X) if there is only one orbit; i.e., for all x, y ∈ X there exists some a ∈ G with ax = y. The triple (G, X, θ) is said to be effective (or G acts effectively on X, or simply, G is effective) if e is the only element of G that fixes every element of X. If G is a subgroup of h(X), the group of self-homeomorphisms of X under function composition, we have a natural action: θ(g, x) = g(x). This action is always effective since any homeomorphism distinct from e must move something. Conversely, if G is effective, a → θa gives a natural group isomorphism from G into h(X). Thus, when we say “G is effective”, we identify G with its image under this isomorphism and thereby view it as a subgroup of h(X). In this way we can consider the commonly used function space topologies on G. It will sometimes happen that θ itself is continuous from the product space G × X to X. In this case, the triple (G, X, θ) will be called a topological transformation group, or equivalently, we say X is a G-space, or again, we will say that the topology on G is admissible, or again, that G acts continuously on X. One special case is noteworthy. The triple (G, G, ·) is always an effective, transitive, topological transformation group, where the action is the group’s multiplication. We will say G is equicontinuous or uniformly equicontinuous, accordingly, if the same is true of the family of maps θa . Thus, G is equicontinuous if for each E ∈ E and each x ∈ X there exists some neighborhood U of x such that (ax, ay) ∈ E for each y ∈ U and each a ∈ G. We say G is uniformly equicontinuous if for each E ∈ E there exists some F ∈ E such that for all (x, y) ∈ F , and all a ∈ G, (ax, ay) ∈ E. Using Δ to denote diagonals, the latter can be rephrased: ΔG · F ⊆ E We make frequent use of such notation, where the multiplication of ordered pairs is coordinatewise. If G is uniformly equicontinuous then it is equicontinuous and each θa is uniformly continuous [18, p. 44]. The two, equicontinuity and uniform equicontinuity, are equivalent when X is compact. When G is effective, the following topologies on G are commonly of interest: τs ⊆ τk ⊆ τu , the pointwise topology, the compact-open topology, and the topology of uniform convergence, respectively. The topology of uniform convergence is the topology that G inherits from the uniformity of uniform convergence with respect

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to E (see [6]). On the other hand, convergence of nets with respect to τk is equivalent to uniform convergence on compact sets. When X is locally compact, there is another important topology, τg , introduced by Arens in [5], that lies between τk and τu . Convergence in τg may be characterized as follows. A net aα converges to −1 a under τg if and only if aα and a−1 , respectively. α converge uniformly on compact subsets of X to a and a We mention some properties of these topologies. Proposition 1.1. Suppose G is effective. (1) If G is equicontinuous, then τs = τk on G. (2) If X is compact, then τk = τg = τu on G. (3) If X is locally compact, then τk is the smallest topology on G making the action θ : G × X → X continuous; however, (G, τk ) is often not a topological group. (4) If X is locally compact, then τg is the smallest topology that makes G into a topological group and is admissible. Proof. (1) and (2) are common results. For (3) and (4) see [4] and [5].

2

We refer the reader to [9] and [11] where properties of τu are investigated. Many of our results require that the maps Tx : G → X given by Tx (g) = gx be almost open; meaning, that for each U open in G and each a ∈ U , Tx (U ) is a neighborhood of Tx (a) in X. Continuous actions with this property have been thoroughly investigated by others. The concept was introduced by Ancel in [1] as weakly micro-transitive actions and has been thoroughly studied under the name d-open actions by Uspenski˘ı in [19] (and elsewhere), Kozlov in [14] and [15], and Kozlov with Chatyrko in [12] and [13]. The requirements for an action to be d-open are fairly modest. For instance, [19, Lemma 9] states that a continuous and transitive action of an ω-narrow group on a Baire space is d-open. In [21], one finds a deeper investigation. For instance, if x ∈ X and G contains an ω-narrow subset S such that Tx (S) is second-category in X, then Tx is almost open. This idea is generalized to higher cardinals. The proof of the result just mentioned (unlike other results in the same paper) doesn’t rely on the action being continuous, but only requires that each θa be a homeomorphism. The following two lemmas give some basic properties of almost open maps. They don’t play a material role in the rest of the article, but they should prove useful in future applications of the main results. These lemmas are surely known. For further results closely related to Lemma 1.2 see Propositions 2 and 3 of [15]. Lemma 1.2. Let f : A → B be a function between topological spaces A and B. Then f is almost open as a function from A to f (A) if and only if it is also almost open as a function from A to f (A). Proof. Let U be open in A and x ∈ U . Assume f : A → f (A) is almost open. So there exists V open relative to f (A) such that f (x) ∈ V ⊆ clf (A) [f (U )]. There is a set W open relative to f (A) such that V = W ∩ f (A). Notice that V is dense in W , so f (x) ∈ W ⊆ clf (A) (V ) ⊆ clf (A) (f (U )) and it follows that f is almost open to f (A). Conversely, assume that f : A → f (A) is almost open. Thus there exists V open relative to f (A) such that f (x) ∈ V ⊆ clf (A) [f (U )]. So V ∩ f (A) is relatively open in f (A) and f (x) ∈ V ∩ f (A) ⊆ clf (A) [f (U )] ∩ f (A) = clf (A) [f (U )] The result follows. 2 Part (1) of the following lemma is essentially [13, Assertion 1]. The converse of part (2) is essentially [15, Lemma 3].

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Lemma 1.3. The following hold for each x ∈ X. ◦

(1) Tx : G → X is almost open if and only if for each U ∈ U(e), x ∈ U x . (2) Suppose G is effective and is (effectively) a topological subgroup of h(X) under some topology. Let G be the closure of G in h(X) and let Tx : G → X denote the natural extension of Tx . If Tx is almost open then Tx is almost open. The converse holds provided Tx is continuous. ◦

Proof. First, assume that for each U ∈ U(e), x ∈ U x . Let V ⊆ G be open and a ∈ V . Then e ∈ a−1 V , so ◦ ◦ ◦ by hypothesis, x ∈ a−1 V x = a−1 (V x ). Thus ax ∈ V x , showing that Tx is almost open. The converse is immediate, establishing (1). Now assume that G is as described in (2). Assume that Tx is almost open. Let U be a neighborhood of ◦ ◦ e in G. Hence U ∩ G is a neighborhood of e in G. Consequently, x ∈ (U ∩ G)x ⊆ U x , and so Tx is almost open by part (1). Now assume that Tx is almost open and continuous. Let U be a neighborhood of e in G. There exists a neighborhood V of e in G such that U = V ∩ G. Since U is dense in V and Tx is continuous, U x is dense ◦ ◦ in V x. We have x ∈ V x = U x , and so Tx is almost open by part (1). 2 For U ∈ U(e) define UL = {(a, b) ∈ G × G | a−1 b ∈ U } UR = {(a, b) ∈ G × G | ba−1 ∈ U } UV = UR ∩ UL = {(a, b) ∈ G × G | b ∈ aU ∩ U a} The families: {UL | U ∈ U(e)}, {UR | U ∈ U(e)}, {UV | U ∈ U(e)} are bases, respectively, of the usual left, right, and two sided uniformities: L, R and V, on G. (See [18].) A uniform space (Y, F) is precompact if for each F ∈ F, Y is a union of finitely many F -small sets; see [6]. In [2], a topological group G is defined to be precompact if for each U ∈ U(e) there exists some finite set A ⊆ G such that G = AU . The two concepts are consistent. It is easily shown that G is precompact in the latter sense if and only if (G, R) (or equivalently, (G, L)) is a precompact uniform space. We will also make use of the Ra˘ıkov completion of G, which we denote ρG. See [2] for an excellent discussion of this topic. 2. Property UE The following definition is the main tool of our investigation. Definition 2.1. A pair of uniformities (D, F) on G and X respectively satisfies property UE if for each F ∈ F, there exists D ∈ D such that D · ΔX ⊆ F . We remark that this is a generalization of a property introduced by J. de Vries in [8]. A topological transformation group (G, X, θ), where (X, E) is a uniform space, is bounded, as defined in that paper, if and only if the pair (R, E) satisfies property UE. Property UE is so named due to the following result. Proposition 2.2. The pair (D, F) satisfies property UE if and only if the family of maps {Tx | x ∈ X}, from (G, D) to (X, F), is uniformly equicontinuous.

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Proof. The family is uniformly equicontinuous if and only if for each F ∈ F there exists some D ∈ D such that (Tx (a), Tx (b)) ∈ F for all (a, b) ∈ D and every x ∈ X [18, Definition 2.27]. However, (Tx (a), Tx (b)) = (ax, bx) = (a, b)(x, x). So the family is equicontinuous if and only if for each F ∈ F there exists some D ∈ D such that DΔX ⊆ F ; that is, if and only if (D, F) satisfies property UE. 2 Property UE is surprisingly strong. When the uniformities are compatible with the topologies, property UE forces the topology on G to be admissible. Theorem 2.3. If there exist uniformities D and E on G and X, respectively, compatible with their topologies, such that the pair (D, E) satisfies property UE, then the topology on G is admissible; i.e., X is a G-space. Proof. By [2, 10.2.8] it suffices to prove that the action is continuous at each point of {e} × X. Let (aα, xα ) be a net in G × X converging to (e, x). Let E[x] be a basic neighborhood of x, where E ∈ E. Let E1 ∈ E such that E1 ◦ E1 ⊆ E. By hypothesis, there exists some D ∈ D such that D · ΔX ⊆ E1 . There exists some α0 such that for all α ≥ α0 , both (e, aα ) ∈ D and (x, xα ) ∈ E1 . Since (e, aα ) ∈ D, (xα , aα xα ) ∈ E1 . Thus (x, aα xα ) ∈ E1 ◦ E1 ⊆ E, showing that aα xα converges to x. 2 Corollary 2.4. If G is effective and (G, τu ) is a topological group, then τu is admissible. Proof. On G, let D denote the uniformity of uniform convergence. The sets DE = {(a, b) ∈ G × G | (a, b)ΔX ⊆ E}, E ∈ E form a base for D. The topology on G induced by this uniformity is τu . Since, DE · ΔX ⊆ E, (D, E) satisfies property UE. The result follows from Theorem 2.3. 2 The following is another interesting side note. Proposition 2.5. If D is a uniformity on G compatible with its topology and we give G × X the product uniformity determined by D and E, then the action θ : G × X → X is uniformly continuous if and only if G is uniformly equicontinuous and the pair (D, E) satisfies property UE. Proof. Assume the action is uniformly continuous. Let E ∈ E. Then there exist D ∈ D and F ∈ E such that D · F ⊆ E. Thus D · ΔX ⊆ E and ΔG · F ⊆ E, showing, respectively, that (D, E) satisfies Property UE and G is uniformly equicontinuous. Conversely, assume G is uniformly equicontinuous and the pair (D, E) satisfies Property UE. Given E ∈ E, choose some E1 ∈ E such that E1 ◦E1 ⊆ E. By property UE there exists some D ∈ D such that D·ΔX ⊆ E1 . Since G is uniformly equicontinuous, there exists some E2 ∈ E such that ΔG · E2 ⊆ E1 . Thus, if (a, b) ∈ D and (x, y) ∈ E2 , we have (ax, ay) ∈ E1 and (ay, by) ∈ E1 . Hence, (ax, by) ∈ E1 ◦ E1 ⊆ E. Thus, the action is uniformly continuous. 2 If U ∈ U(e), we use (e, U ) to denote {e} × U . Lemma 2.6. (1) For each U ∈ U(e), UR ΔX = (e, U )ΔX ⊆ UL ΔX . (2) For each U ∈ U(e), UL ΔX = ΔG UR ΔX . (3) If (L, E) satisfies property UE then so does (R, E).

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Proof. Since UR = (e, U )ΔG , UR ΔX = (e, U )ΔG ΔX = (e, U )ΔX . Also, (e, U ) ⊆ UL . Thus (1) holds. From (1) and the fact that UL is left invariant, we have ΔG UR ΔX ⊆ ΔG UL ΔX = UL ΔX . On the other hand, each element of UL ΔX can be expressed in the form (ax, aux), where a ∈ G, u ∈ U , and x ∈ X. However (ax, aux) = (a, a)(e, u)(x, x) ∈ ΔG UR ΔX , proving (2). (3) follows from (1). 2 Recall that if X is compact, there is a unique uniformity compatible with its topology, which we will denote as EC ; see [20, Theorem 36.19]. Theorem 2.7. If X is compact then the following statements are equivalent. (1) There exists a uniformity D on G compatible with its topology such that the pair (D, EC ) satisfies property UE. (2) The topology on G is admissible. (3) The pair (R, EC ) satisfies property UE. Proof. (1) =⇒ (2) is given by Theorem 2.3. Now assume that the topology on G is admissible. Let E ∈ EC . Then there exists some symmetric F ∈ EC such that F ◦ F ⊆ E. By the continuity of the action, if x ∈ X, there exists a neighborhood V of x in X and some U ∈ U(e) such that U V ⊆ F [x]. We will first show that for each y ∈ V , U y ⊆ E[y]. Let u ∈ U and y ∈ V . Since, e ∈ U and U V ⊆ F [x], y and uy both lie in F [x]. Since F is symmetric, it follows that (y, uy) ∈ F ◦ F ⊆ E, showing uy ∈ E[y]. This completes the proof that U y ⊆ E[y] for each y ∈V. Now, since X is compact it can be covered by finitely many such Vi , each with its corresponding Ui . Let U be the intersection of the Ui . Each y ∈ X lies in some Vi , so U y ⊆ E[y]. Thus {y} × U y ⊆ E for each y ∈ X, which in turn implies that (e, U )ΔX ⊆ E. Since (e, U )ΔX = UR ΔX by Lemma 2.6, (R, EC ) has property UE, completing the proof that (2) =⇒ (3). (3) =⇒ (1) is of course immediate. 2 We now turn our attention to showing that if X is a G-space and each Tx is almost open, then there is a uniformity E  on X, compatible with its topology, such that (R, E  ) satisfies property UE. This uniformity ˜G in [12] (proven equal is not new. Kozlov and Chatyrko studied the same uniformity, called U G in [7] and U in [12]), both defined in terms of covering uniformities. In [12, Theorem 3], they show that this uniformity is compatible with the topology on X, and, (under the terminology of de Vries) the pair (R, E ) satisfies property UE (and many other properties as well). Thus, nearly all of Theorem 2.9 given below must be attributed to them. We offer an alternate presentation in terms of diagonal uniformities. We begin with a lemma. A related result can be found as [13, Lemma 3]. Lemma 2.8. Suppose X is a G-space and V is a symmetric neighborhood of e in G. (1) If each Tx is almost open, then for all x, y ∈ X such that y ∈ V x, x ∈ V 2 y. (2) If each Tx is open, then for each x ∈ X, V x ⊆ V 2 x. Proof. Let y ∈ V x. If Ty is almost open, V y is a neighborhood of y. Hence V y ∩ V x = ∅. Thus there exists a net vα y converging to some vx, where vα , v ∈ V . Therefore, v −1 vα y converges to x, showing that x ∈ V 2 y. This establishes (1). For (2), assume that each Tx is open and let x ∈ X. Let y ∈ V x. Since V y is a neighborhood of y, V y meets V x. Thus there exist v, w ∈ V such that vy = wx. Hence y ∈ V 2 x, establishing (2). 2

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The following theorem may be viewed as a generalization of the fact that the quotient uniformity on a coset space G/H induced by R is compatible with the quotient topology [18, Theorem 5.21]. Theorem 2.9 (Kozlov, Chatyrko). If X is a G-space and with weakly micro-transitive action, then the sets UR defined by UR = {(x, y) ∈ X × X | y ∈ U x}, U ∈ U(e) form a base for a uniformity E  on X, compatible with its topology, such that the pair (R, E  ) satisfies property UE. Furthermore, if U ∈ U(e) is invariant, then for each g ∈ G, (g, g) · UR = UR . Proof. Clearly ΔX ⊆ UR . If U, V ∈ U(e), then (U ∩ V )R ⊆ UR ∩ VR , establishing one requirement to be a base for a uniformity. Given U ∈ U(e), choose some V ∈ U(e) such that V 2 ⊆ U . Now say (x, y) and (y, z) are in VR . Then y ∈ V x and z ∈ V y. We have z ∈ V · (V x) ⊆ V 2 x = V 2 x ⊆ U x Consequently, (x, z) ∈ UR , showing that VR ◦ VR ⊆ UR . Finally, given U ∈ U(e), let V be a symmetric neighborhood of e such that V 3 ⊆ U . Let (x, y) ∈ VR . Then y ∈ V x. By Lemma 2.8, x ∈ V 2 y. Hence x ∈ V x ⊆ (V · V 2 y) ⊆ (V 3 y) ⊆ U y Thus, (y, x) ∈ UR showing that (VR )−1 ⊆ UR . This completes the proof that the sets UR form base for a uniformity E  on X. Notice that UR [x] = U x, which is a neighborhood of x since Tx is almost open. On the other hand, if W is an open set in X and x ∈ W , there exists some W1 open in X such that x ∈ W1 ⊆ W1 ⊆ W (uniform spaces are regular). There also exists some U ∈ U(e) such that U x ⊆ W1 . We have UR [x] = U x ⊆ W . This establishes that E  is compatible with the topology on X. Now, if (a, b) ∈ UR and x ∈ X, then b ∈ U a, so bx ∈ U ax ⊆ U ax. Thus (ax, bx) ∈ UR , showing UR · ΔX ⊆ UR and property UE holds for the pair (R, E  ). Finally, let U be an invariant neighborhood of e, (x, y) ∈ UR , and g ∈ G. Then y ∈ U x, so gy ∈ gU x = gU x = U gx Hence, (gx, gy) ∈ UR , showing that (g, g) · UR ⊆ UR . However, it follows that the two sets are equal, for if (x, y) ∈ UR , then (x, y) = (gg −1 x, gg −1 y) = (g, g) · (g −1 x, g −1 y) ∈ (g, g) · UR

2

Throughout the sequel, when X is a G-space with weakly micro-transitive action, we shall let E  denote the uniformity in Theorem 2.9, and τu the topology of uniform convergence on G determined by E  . Corollary 2.10. If each map Tx is almost open, then the following statements are equivalent. (1) There exist uniformities D and E on G and X, compatible with their topologies, such that the pair (D, E) satisfies property UE. (2) The topology on G is admissible.

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(3) There exists a uniformity F on X compatible with its topology such that the pair (R, F) satisfies property UE. Proof. By Theorem 2.3, (1) implies (2). By Theorem 2.9, (2) implies (3). The remaining implication is immediate. 2 The following known fact is distinguished as a lemma, only because we will need it later. Lemma 2.11. With respect to the action of G on itself by multiplication, the pair (R, R) always satisfies property UE. Proof. This is immediate since UR ΔG = UR .

2

3. The SIN property We say G is a SIN group, or balanced, if it has a neighborhood base (equivalently, open neighborhood base) at e of invariant neighborhoods. Here are some common properties of SIN groups. Proposition 3.1. The following all hold for any topological group G. G is SIN if and only if for each U ∈ U(e) there exists V ∈ U(e) such that aV a−1 ⊆ U for each a ∈ G. Subgroups of SIN groups are SIN. All abelian groups are SIN. All precompact topological groups are SIN [2, Corollary 3.7.8]. If H is a subgroup of G, then H is SIN if and only if H is SIN. G is SIN if and only if L = R. G is SIN if and only if whenever (aα , bα ) is a net in G × G such that aα bα converges to e, then bα aα also converges to e. (8) Among metric groups, the SIN-groups are precisely those for which there exists an invariant metric.

(1) (2) (3) (4) (5) (6) (7)

The following theorem is our characterization of SIN groups as acting groups. Theorem 3.2. A topological group H is a SIN group if and only if there is a uniform space (Y, F) such that H acts effectively on Y , the pair (L, F) has property UE, and the topology on H is contained in τu (as determined by F). Proof. To prove sufficiency, let (aα , bα ) be a net in H × H such that aα bα converges to e. To conclude that H is SIN, it suffices to prove that bα aα also converges to e. Let F ∈ F. By hypothesis there exists some U ∈ U(e) such that UL · ΔY ⊆ F . There exists an α0 such that for α ≥ α0 , aα bα ∈ U , and thus −1 (e, aα bα ) ∈ UL . This in turn implies that (bα , bα aα bα ) ∈ UL . If y ∈ Y , then (b−1 α y, bα y) ∈ ΔY . Hence, −1 (bα , bα aα bα )(b−1 α y, bα y) = (y, bα aα y) ∈ F

Thus bα aα converges uniformly to e. Since the topology of H is contained in τu , bα aα converges to e in the topology of H. Now assume that H is SIN. We will let (H, R) take the role of (Y, F). Of course, H acts effectively on itself under multiplication. Since H is SIN, L = R, so by Lemma 2.11, the pair (L, R) satisfies property UE.

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It remains only to be shown that the topology of H is contained in τu , as determined by R. Trivially, if aα is a net in H such that aα converges uniformly to an element a with respect to R, then aα converges to a in the topology of H. Hence, the topology of H is contained in τu . 2 From Theorems 2.7 and 3.2, we immediately have: Theorem 3.3. Let X be a compact G-space with effective action, where G is given the compact open topology. Then G is a SIN group if and only if (L, EC ) has property UE. Theorem 3.4. Let X be G-space with effective, weakly micro-transitive action, where the topology of G is contained in  τu . Then G is a SIN group if and only if and only if (L, E  ) has property UE. Proof. By Theorem 2.9, (R, E  ) has property UE. Thus, the forward implication follows from the fact that if G is SIN then L = R. The converse follows from Theorem 3.2. 2 4. Equicontinuity Theorem 4.1. Let X be a G-space with weakly micro-transitive action, and E any uniformity on X compatible with its topology. If the pair (L, E) satisfies property UE, then G is equicontinuous with respect to E. Proof. Let E ∈ E and x ∈ X. Let E1 be a symmetric element of E such that E1 ◦ E1 ⊆ E. Since (L, E) has property UE, there is some U ∈ U(e) such that UL · ΔX ⊆ E1 . We may assume that U is symmetric. Since ◦ Tx is almost open, x ∈ U x . Thus, there exists some V open in X such that x ∈ V ⊆ U x. We will show that V is the required neighborhood of x. Let y ∈ V and a ∈ G. Then ay ∈ aV ⊆ aU x = aU x. Since aV is open in X, there exists some E2 ∈ E such that E2 ⊆ E1 and E2 [ay] ⊆ aV . Since E2 [ay] is a neighborhood of ay, E2 [ay] must meet aU x. Thus there exists some u ∈ U such that aux ∈ E2 [ay]. Thus (ay, aux) ∈ E2 ⊆ E1 . Since U is symmetric, (au, a) ∈ UL and (aux, ax) ∈ UL · ΔX ⊆ E1 . Thus, E1 being symmetric, (ax, ay) ∈ E1 ◦ E1 ⊆ E. 2 Corollary 4.2. If X is a compact G-space, G is SIN, and the action is weakly micro-transitive, then G is uniformly equicontinuous. Proof. By Theorem 2.7, (R, EC ) has property UE. Since G is SIN, (L, EC ) has property UE, and G is equicontinuous by Theorem 4.1. Since X is compact, G is uniformly equicontinuous. 2 Theorem 4.3. If G is a SIN group, X is a G-space, and the action is weakly micro-transitive, then G is uniformly equicontinuous with respect to E  . Proof. If G is SIN, then the sets UR where U is invariant form a base for E  . According to Theorem 2.9, if U is invariant, ΔG UR = UR . It follows that G is uniformly equicontinuous with respect to E  . 2 It is interesting to note that when G is SIN, the sets UR where U is invariant form a base for E  , and by Theorem 2.9, these sets are invariant with respect to translations by elements (g, g). This is a generalization of the metric case, where it can be shown that if G is uniformly equicontinuous, X has a metric making each θa an isometry [5, Corollary, p. 604]. Theorem 4.4. If G is uniformly equicontinuous, then (L, E) satisfies property UE if and only if (R, E) satisfies property UE.

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Proof. Assume (R, E) has property UE. Let E ∈ E. Since G is uniformly equicontinuous there exists some E1 ∈ E such that ΔG E1 ⊆ E. Since (R, E) has property UE, there exists U ∈ U(e) such that UR ΔX ⊆ E1 . Thus, by part (2) of Lemma 2.6, UL ΔX = ΔG UR ΔX ⊆ ΔG E1 ⊆ E. Hence (L, E) satisfies property UE. The converse follows from part (3) of Lemma 2.6. 2 Corollary 4.5. If X is a G-space with effective, weakly micro-transitive action, and the topology of G is contained in τu , then G is a SIN group if and only if G is uniformly equicontinuous with respect E  . Proof. The forward implication follows from Theorem 4.3. Now assume G is uniformly equicontinuous with respect to E  . According to Theorem 2.9, (R, E  ) has property UE. By Theorem 4.4, (L, E  ) has property UE. Thus G is SIN by Theorem 3.2. 2 Corollary 4.6. If X is a compact G-space with effective, weakly micro-transitive action, where G is given the compact-open topology, then the following statements are equivalent. (1) G is uniformly equicontinuous. (2) G is a SIN group. (3) (L, EC ) satisfies property UE. Proof. Since X is compact, the uniformity E  given in Theorem 2.9 equals EC . Thus (1) and (2) are equivalent by Corollary 4.5. Also, (2) is equivalent to (3) by Theorem 3.4. 2 5. Compactness In this section, we address compactness or local compactness of G. Remark 1. If X is locally compact, then (h(X), τg ) is always complete with respect to the two-sided uniformity [5, Theorem 6], and hence Ra˘ıkov complete [2, Theorem 3.6.25]. So if G is topologized with τg when X is locally compact, or with τk = τg when X is compact, then ρG, the Ra˘ıkov completion of G, is topologically isomorphic to G, the closure of G in (h(X), τg ) [2, Theorem 3.6.14]. We state two classical results. Proposition 5.1. Every equicontinuous group of homeomorphisms of a compact space is precompact under the compact open topology. Proof. Assume G is equicontinuous and X is compact. Since X is compact, τk = τu . Let W denote the uniformity of uniform convergence as determined by EC . By Ascoli’s Theorem [6, X, 2.5], G is precompact with respect to W. However, W coincides with the right uniformity on the group (G, τu ) ([18, Exercise 2 Chapter 2]). Since (G, τu ) = (G, τk ), G is precompact with respect to R, and so is precompact. (See comments at end of preliminary section.) 2 Proposition 5.2. [5, Theorem 7] If X is connected and locally compact, G has the topology τg , and the action is effective and uniformly equicontinuous, then ρG is locally compact. Theorem 5.3. If X is a compact G-space with effective, weakly micro-transitive action, where G given the compact-open topology, then the following statements are equivalent. (1) The pair (L, EC ) satisfies Property UE.

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(2) (3) (4) (5) (6)

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G is equicontinuous. G is uniformly equicontinuous. G is precompact. ρG is compact. G is a SIN-group.

Proof. The equivalence of (1), (3) and (6) is given by Corollary 4.6. The equivalence of (2) and (3) follows from the compactness of X. By Proposition 5.1, (3) implies (4). The implication (4) =⇒ (5) is a standard result ([2, Theorem 3.7.16]). Compact groups are SIN and subgroups of SIN groups are SIN, so (5) =⇒ (6). 2 Theorem 5.4. Suppose X is a connected, locally compact G-space, where G is effective and weakly microtransitive under the topology τg . If G is a SIN group, then ρG is locally compact. Proof. By Theorem 4.3, G is uniformly equicontinuous with respect to the uniformity E  . The result now follows from Proposition 5.2. 2 We conclude with a proof of the theorem stated in the introduction. Proof. The implications (3) ⇐⇒ (5) ⇐⇒ (6) ⇐⇒ (7) follow from Theorem 5.3. The implication (1) =⇒ (2) is immediate. Now assume (2) holds. Let D and E be the corresponding metric uniformities on G and X, respectively. Then from (2), (D, E) satisfies property UE. However, since d is left-invariant, D = L. Thus G is equicontinuous by Theorem 4.1, and so (2) =⇒ (3). If (3) holds, G is uniformly equicontinuous since X is compact. Then by [5, Corollary, p. 604], X has a metric D compatible with its topology making each element of G an isometry. Hence (3) =⇒ (4). The equivalence of (4) and (5) is a classical result of Eilenberg [17], and in fact, places no requirements on Tx . Now assume (7) holds. Then by the established implications, (5) also holds, and by the equivalence of (4) and (5) applied to G, X has a metric D making each element of G an isometry. Let d be the ordinary sup-metric on G determined by D. Then d is right-invariant. If a, b, c ∈ G, then d(ca, cb) = sup{D(cax, cbx) | x ∈ X}. However, D(cax, cbx) = D(ax, bx), so d(ca, cb) = d(a, b). Thus d is invariant, and d and D satisfy the condition in (1). Hence (8) holds. The implication (8) =⇒ (1) is trivial. 2 References [1] Fredric D. Ancel, An alternative proof and applications of a theorem of E.G. Effros, Mich. Math. J. 34 (1) (1987) 39–55. [2] A. Arhangel’skii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Scientific Publishing Co. Pte. Ltd., Paris, Hackensack, NJ, ISBN 978-90-78677-06-2, 2008, xiv+781 pp. [3] A.V. Arhangel’skii, J. van Mill, Topological homogeneity, in: K.P. Hart, J. van Mill, P. Simon (Eds.), Recent Progress in General Topology III, Atlantis Press, Paris, ISBN 978-94-6239-023-2, 2014, 978-94-6239-024-9, viii+903 pp. [4] R.F. Arens, A topology for spaces of transformations, Ann. Math. (2) 47 (1946) 480–495. [5] R.F. Arens, Topologies for homeomorphism groups, Am. J. Math. 68 (1946) 593–610. [6] N. Bourbaki, General Topology, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, ISBN 3-540-19372-3, 1998, Chapters 5–10. Translated from the French. Reprint of the 1989 English translation, iv+363 pp. [7] V.A. Chatyrko, K.L. Kozlov, The maximal G-compactifications of G-spaces with special actions, in: Proceedings of the Ninth Prague Topological Symposium, 2001, Topol. Atlas, North Bay, ON, 2002, pp. 15–21. [8] Jan de Vries, On the existence of G-compactifications, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 26 (3) (1978) 275–280. [9] Jean Dieudonné, On topological groups of homeomorphisms, Am. J. Math. 70 (1948) 659–680.

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