On proximal fineness of topological groups in their right uniformity
aa r X i v : . [ m a t h . GN ] A p r ON PROXIMAL FINENESS OF TOPOLOGICAL GROUPS INTHEIR RIGHT UNIFORMITY
A. BOUZIAD
Abstract.
A uniform space X is said to be proximally fine if every proximallycontinuous map on X into a uniform is uniformly continuous. We supplya proof that every topological group which is functionnaly generated by itsprecompact subsets is proximally fine with respect to its right uniformity. Onthe other hand, we show that there are various permutation groups G on theintegers N that are not proximally fine with respect to the topology generatedby the sets { g ∈ G : g ( A ) ⊂ B } , A, B ⊂ N . Introduction
A function f : X → Y between tow uniform spaces is said to be proximallycontinuous if for every bounded uniformly continuous function g : Y → R , thecomposition function g ◦ f : X → R is uniformly continuous; the reals R beingequipped with the usual metric. A uniform space X is said to be proximally fine if every proximally continuous function defined on X is uniformly continuous. Itis well-known that metric spaces and all products of metric uniform spaces areproximally fine; however, there are many uniform spaces that are not proximallyfine although they are topologically well-behaved (some may be locally compactand even discrete). We refer the reader to Huˇsek’s recent paper [12] for more in-formation about proximally fineness of general uniform spaces. We are interestedhere in the fine proximal condition of the topological groups when these spacesare endowed with their right uniformity. This subject seems to have no specificliterature, although there are some questions which could have been naturallyaddressed in this sitting, such as the Itzkowitz problem for metric and/or locallycompact groups [13] (the link between Itzkowitz’s problem and proximity theorywas made later in [6]).In view of the close relationship between the right uniformity of topologicalgroups and the system of neighborhoods of their identity, it is reasonable to expectthat the proximal fineness of a given topological group G is satisfied provided that Mathematics Subject Classification. a not too much restrictive condition is imposed on the topology of G . This feelingis heightened by Corollary 2.6 in this note asserting that it is sufficient to assumethat G is functionally generated (in Arkhangel’skiˇı’s sense) by its precompactsubsets. In fact, this is still true under much less restrictive conditions (Theorem2.4 below). Let us mention that one part of Theorem 2.4 was asserted withoutproof in [5] . The main subject of this note can be examined in the context of G -sets (or G -spaces) without significantly altering its essence, so the main resultis stated and established in a somewhat more general form (Theorem 2.5).Some examples of non-proximally fine groups are given in Section 3. To dothat, we consider the topology τ on N N of uniform convergence when (the targetset) N is endowed with the Samuel uniformity. We show in Corollary 3.4 thatfor any permutation group H on N for which all but finitely many orbits arefinite and uniformly bounded, the only group topology on H which finer than τ and proximally fine is the discrete topology. The question of whether there is anabelian (or at least SIN) group which is non-proximally fine group is left open.2. Main result
For undefined terms we refer to the books [8] and [15]. One of the tools usedhere is Katetov’s extension theorem of uniformly continuous bounded real-valuedfunctions [14]. The following statement is also well-known (see[8]); its proof isoutlined here for the sake of completeness and because it can be adapted to showa result in the same spirit that will be used in the proof Theorem 2.5. As usual, if( X, U ) is a uniform space, where U is the uniform neighborhoods of the diagonalsof X , then for A ⊂ X and U ∈ U , U [ A ] stands for the set of y ∈ X such that( x, y ) ∈ U for some x ∈ A . Proposition 2.1.
Let f : X → Y , where ( X, U ) and ( Y, V ) are two uniformspaces. Then the following are equivalent: (1) f : X → Y is proximally continuous, (2) for every A ⊂ X and V ∈ V , there exists U ∈ U such that f ( U [ A ]) ⊂ V [ f ( A )] .Proof. To show that 1) implies 2), let d be a bounded uniformly continuouspseudometric on Y such that ( x, y ) ∈ V whenever d ( x, y ) < The author is indebted to Professor Michael D. Rice for his interest in this result from [5]and its proof, which motivated the present work.
N PROXIMAL FINENESS OF TOPOLOGICAL GROUPS 3 the uniformly continuous function φ on Y defined by φ ( y ) = d ( y, f ( A )). Since φ ◦ f is uniformly continuous, there is U ∈ U such that ( x, y ) ∈ U implies | φ ( f ( x )) , φ ( f ( y )) | <
1. Then f ( U [ A ]) ⊂ V [ f ( A )].For the converse, let φ : Y → R be a bounded uniformly continuous functionand let ε >
0. Define V = { ( x, y ) ∈ Y × Y : | φ ( x ) − φ ( y ) | < ε/ } . Then V ∈ V and since φ is bounded, there is a finite set F ⊂ Y so that Y = V [ F ]. Let A z = f − ( W [ z ]), z ∈ F , and choose U ∈ U such that f ( U [ A z ]) ⊂ W [ f ( A z )] foreach z ∈ F . Then | φ ◦ f ( x ) − φ ◦ f ( y ) | ≤ ε for each ( x, y ) ∈ U . (cid:3) Let us say that a topological group G is proximally fine if the uniform space( G, U r ) is proximally fine, where U r is the right uniformity of G . It is equivalentto say that ( G, U l ) is proximally fine, where U l is the left uniformity of G . Recallthat a basis of U r (respectively, U l ) is given by the sets of the form { ( g, h ) ∈ G × G : gh − ∈ V } (respectively, { ( g, h ) ∈ G × G : g − h ∈ V } ) as V runs overthe set V ( e ) of neighborhoods of the unit e of G . Proposition 2.2.
Let G be a topological group, ( Y, U ) a uniform space and let f : G → Y be a function. For g ∈ G , let ψ g : G → Y be the function defined by ψ g ( h ) = f ( gh ) . Then, the following are equivalent: (1) f is uniformly continuous, G being equipped with the right uniformity, (2) the function ψ : g ∈ G → ψ g ∈ Y G is continuous when Y G is endowedwith the uniformity of uniform convergence on G .Proof. To show that (1) implies (2), suppose that f is right uniformly continuousand let U ∈ U . There is a neighborhood V of e such that xy − ∈ V implies( f ( x ) , f ( y )) ∈ U . Let g ∈ G . Then V g is a neighborhood of g in G and for each h ∈ V g and x ∈ G we have hx ( gx ) − ∈ V hence ( f ( hx ) , f ( gx )) ∈ U , that is,( ψ h ( x ) , ψ g ( x )) ∈ U .To show that (2) implies (1), let U ∈ U and choose V ∈ V ( e ) such that( ψ g ( x ) , ψ e ( x )) ∈ U for every g ∈ V and x ∈ G . Then, for every g, h ∈ G such that h ∈ V g we have ( ψ hg − ( x ) , ψ e ( x )) ∈ U for each x ∈ G , equivalently,( f ( hx ) , f ( gx )) ∈ U , for every x ∈ G . (cid:3) It is possible to expand substantially the framework of the starting topic of thisnote without major changes as follows: Let G be a topological group and let X be a G -set, that is, X is a nonempty set for which there is a map ∗ : G × X → X satisfying ( gh ) ∗ x = g ∗ ( h ∗ x ) for every g, h ∈ G and x ∈ X . The function ∗ is A. BOUZIAD called a left action of G on X . To simplify, write gx in place of g ∗ x and U A inplace of U ∗ A if U ⊂ G and A ⊂ X . No topology will be required on X . Let( Y, V ) be a uniform space and f : X → Y a function. It is consistent with thedefinitions given above to say that a f is right uniformly continuous if for each V ∈ V , there is U ∈ V ( e ) such that ( f ( gx ) , f ( hx )) ∈ V for each x ∈ X , whenever gh − ∈ U . Similarly, the function f is said to be right proximally continuous iffor each bounded uniformly continuous function φ : Y → R , the function φ ◦ f isright uniformly continuous. It is easy to check (see the proof of Proposition 2.2)that f is right uniformly continuous iff the function ψ : g ∈ G → f ( gx ) ∈ Y X is continuous when Y X is endowed with the uniform convergence. Similarly, asimple adaptation of the proof of Proposition 2.1 shows that f is right proximallycontinuous iff for each A ⊂ GX and V ∈ V , there is U ∈ V ( e ) such that f ( U A ) ⊂ V [ f ( A )].The following properties are required for Theorem 2.5; we are formulating themseparately to reduce the proof to its essential components. Let X be a G -set and f : X → Y a right proximally continuous, as defined above. Then, for each A ⊂ G :(c1) for every x ∈ X , the function g ∈ G → f ( gx ) ∈ Y is continuous,(c2) if ψ | A is right uniformly continuous, then ψ | A is right uniformly continuous,(c3) if a ∈ A is a point of continuity of ψ | A , then a is a point of continuity of ψ | A .We check the validity of these properties for the benefit of the reader. Let V ∈ V . For every x ∈ X and g ∈ G , there is U ∈ V ( e ) such that f ( V gx ) ⊂ V [ f ( gx )]. Since V g is a neighborhood of g in G , (c1) holds. Property (c2) and(c3) follows from (c1). For, if x ∈ X , U ∈ V ( e ) are such that ( f ( ax ) , f ( bx )) ∈ V for every a, b ∈ A with a ∈ U U b , then (c1) implies that f ( gx ) , f ( gx )) ∈ V foreach g, h ∈ A such that g ∈ U h . Taking x arbitrary in X gives (c2). For (c3),let U be an open neighborhood of the unit in G such that ( f ( gx ) , f ( ax )) ∈ V forever g ∈ U a ∩ A and x ∈ X . Since U a ∩ A ⊂ U a ∩ A , it follows from (c1) thatfor every g ∈ U a ∩ A and x ∈ X , ( f ( gx ) , f ( ax )) ∈ V .To establish Theorem 2.5 we also need the next key lemma; this is a well-knowntool in the theory of proximity spaces (most often with W instead of W ). Lemma 2.3.
Let X be a set and W be a symmetric binary relation on X . Then,for every infinite cardinal η and for every sequence ( x n , y n ) n<η ⊂ X × X such that N PROXIMAL FINENESS OF TOPOLOGICAL GROUPS 5 ( x n , y n ) W for each n < η , there is a cofinal set A ⊂ η such that ( x n , y m ) W for every n, m ∈ A .Proof. Replacing η by its cofinality cf( η ), we may suppose that η is regular. Let M ⊂ η be a maximal set satisfying ( x n , y m ) W for every n, m ∈ M . If M is cofinal in η , the proof is finished, so suppose that M is not cofinal in η . Foreach j ∈ M , let A j = { n < η : ( x n , y j ) ∈ W } , B j = { n < η : ( x j , y n ) ∈ W } , C j = A j ∪ B j and C = ∪ j ∈ M C j . The maximality of M implies that η ⊂ M ∪ C .Since η is regular, there is j ∈ M such that C j is cofinal in η , therefore A j or B j is cofinal in η . We suppose that it is A j , the other case is similar. Let n, m ∈ A j .Then ( x n , y j ) ∈ W and ( x m , y j ) ∈ W , hence ( x n , y m ) W since ( x m , y m ) W (recall that W is symmetric). Similarly, ( x m , y n ) W . (cid:3) We will now specify a few topological concepts that will be used in what follows.The first is a variant of Herrlich’s notion of radial spaces [11]. Radial spaces werecharacterized by A.V. Arhangel’skiˇı [3] as follows: A space X is radial if andonly if for each x ∈ X and A ⊂ X such that x ∈ A , there is B ⊂ A of regularcardinality | B | such that x ∈ C for every C ⊂ B having the same cardinalityas B . Let us say that a subset A of X is relatively o-radial in X if for everycollection ( O i ) i ∈ I of open sets in X and x ∈ A such that x ∈ ∪ i ∈ I O i ∩ A \ ∪ i ∈ I O j , there is a set J ⊂ I of regular cardinality such that x ∈ ∪ j ∈ L O j whenever L ⊂ J and | L | = | J | . If the set J can always be chosen countable, then A is said to be relatively o-Malykhin in X . All closures are taken in X .Every almost metrizable (in particular, ˇCech-Complete) group is o-Malykhin(in itself). More generally, every inframetrizable group [15] is o-Malykhin, see [6].Following Arhangel’skiˇı [2], the space X is said to be strongly functionallygenerated (respectively, functionally generated ) by a collection M of subsets of X if for every discontinuous function f : X → R , there exists A ∈ M such that therestriction f | A : A → R of f to the subspace A of X is discontinuous (respectively,has no continuous extension to X ).The following is the main result of this note. The statement corresponding tothe case (2) was asserted (without proof) in [5]. Theorem 2.4.
Let G be a topological group satisfying at least one of the following: A. BOUZIAD (1) G is functionally generated by the sets A ⊂ G such that AA − is relativelyo-radial in G , (2) G is strongly functionally generated by the sets A ⊂ G such that A isrelatively o-radial in G .Then G is proximally fine. According to Proposition 2.2 and keeping the above notations, Theorem 2.4 isobtained from the following general result by considering the left action of G onitself. Theorem 2.5.
Let G be a topological group and suppose that for each discontin-uous bounded function α : G → R , there is a set A ⊂ G having at least one ofthe following conditions: (1) α | A has no continuous extension to G and AA − is relatively o-radial in G , (2) α | A is discontinuous at some point of A and A is relatively o-radial in G .Let X be a G -set, ( Y, V ) a uniform space and let f : X → Y be a right proximallycontinuous. Then f : G → X is right uniformly continuous.Proof. We have to show that the function ψ : g ∈ G → ψ ( g ) ∈ Y X (where ψ ( g )( x ) = f ( gx )) is continuous. We proceed by contradiction by supposing that ψ is not continuous. Then, there is a bounded uniformly continuous function θ : Y X → R such that θ ◦ ψ is not continuous (see [8]). Let A ⊂ G satisfyingat least one of the conditions (1) and (2) with respect to the function θ ◦ ψ . Incase (1), there is no compatible uniformity on G making uniformly continuousthe function θ ◦ ψ | A ; for, otherwise, Katetov’s theorem would give us a continuousextension of θ ◦ ψ | A . In particular, ψ | A is not right uniformly continuous. Asremarked above, it follows from (c2) that ψ | A is not right uniformly continuous.There is then an open and symmetric W ∈ U such that for every V ∈ V ( e ), thereexist a V ∈ V , g V ∈ A and x V ∈ X satisfying a V g V ∈ A and( f ( a V g V x V ) , f ( g V x V )) W . (2.1)For each V ∈ V ( e ), let h V = g V x V and define O V = { g ∈ G : ( f ( gh V ) , f ( a V h V )) ∈ W } . Since the functions g ∈ G → f ( gh V ) ∈ Y , V ∈ V ( e ), are continuous (c1), each O V is open in G . Since a V ∈ O V ∩ Ag − V ⊂ AA − and a V ∈ V for each V ∈ V ( e ), N PROXIMAL FINENESS OF TOPOLOGICAL GROUPS 7 it follows that e ∈ [ V ∈V ( e ) O V ∩ ( AA − ) . (2.2)We also have e O V , for each V ∈ V ( e ). Indeed, otherwise, there exists g ∈ O V such that ( f ( gh V ) , f ( h V )) ∈ W , hence ( f ( a V h V ) , f ( h V )) ∈ W which contradicts(2.1). Since AA − is relatively o-radial in G , in view of (2.2), there is a setΓ ⊂ V ( e ) of regular cardinal such that for each set I ⊂ Γ of the same cardinal asΓ, we have e ∈ [ V ∈ I { g ∈ G : ( f ( gh V ) , f ( a V h V )) ∈ W } . (2.3)By Lemma 2.3 and (2.1), there is I ⊂ Γ such that | I | = | Γ | (since | Γ | is regular)and ( f ( a U h U ) , f ( h V )) W for every U, V ∈ I . Since f is right proximallycontinuous, there exists V ∈ V ( e ) such that f ( V { h U : U ∈ I } ) ⊂ W [ f ( { h U : U ∈ I } )] . (2.4)By (2.3) applied to I , there is U ∈ I such that V ∩ O U = ∅ . Let g ∈ V be such that ( f ( gh U ) , f ( a U h U )) ∈ W and by (2.4) let U ∈ I be so that( f ( gh U ) , f ( h U )) ∈ W . It follows that ( f ( a U h U ) , f ( h U )) ∈ W , which is acontradiction. Therefore, ψ is continuous in case (1).In case (2), A is o-radial in G and θ ◦ ψ | A is discontinuous at some point a ∈ A .Since θ is continuous, ψ | A is necessarily discontinuous at a . It follows from theproperty (c3) that ψ | A is discontinuous at a . Let W ∈ U be symmetric and opensuch that for every V ∈ V ( e ), there exist a V ∈ V and x V ∈ G , such that a V a ∈ A and ( f ( a V ax V ) , f ( ax V )) W . Taking h V = ax V for each V ∈ V ( e ), we have e ∈ [ V ∈V ( e ) { g ∈ G : ( f ( gh V ) , f ( h V )) ∈ W } ∩ ( Aa − ) . It is easy to see that Aa − is relatively o-radial in G , therefore the proof can becontinued and concluded in the same way as in the first case. It should be notedthat Katetov’s theorem was not used in this case. (cid:3) Let A ⊂ G , where G is a topological group. It is proved in [6] that AA − isrelatively o-Malykhin in G , provided that A is left and right precompact. ThusTheorem 2.4 yields: Corollary 2.6.
Every topological group G which is functionally generated by thecollection of its precompact subsets is proximally fine. A. BOUZIAD
In view of the role played by locally compact groups in many areas of mathe-matics, it is worth mentioning the following particular case of Corollary 2.6.
Corollary 2.7.
Every locally compact topological group is proximally fine.
Recall that the group G is said to SIN (or with small invariant neighborhoodsof the identity) if its left uniformity and right uniformity are equal. The group G is said to be FSIN (or functionally balanced) if every bounded left uniformlycontinuous function f : G → R is right uniformly continuous. The group G issaid to be strongly FSIN if every every real-valued uniformly continuous functionon G is left uniformly continuous. The question whether every FSIN group isSIN is called Itzkowits problem and is still open. We refer the reader to [5] formore information; see also [16] for a very recent contribution to this topic. Thecorollary of Theorem 2.4 that every FSIN group is SIN provided that it is stronglyfunctionally generated by its relatively o-radial subsets has already been stated(implicitly and without proof) in [5]. This is supplemented by the following: Corollary 2.8.
Every FSIN group G which is functionally generated by the sets A ⊂ G such that AA − is relatively o-radial in G is a SIN group. To conclude this section, we would like to take this opportunity to comment onthe parenthesized question of [5, Question 6] whether every bounded topologicalgroup is FSIN. The answer is of course no, since FSIN is a hereditary property(by Katetov’s theorem) and every group is isomorphic both algebraically andtopologically to a subgroup of a bounded group [10] (see also [9]).3.
Examples
In this section, we give some examples of non-proximally fine Hausdorff topo-logical groups and examine their behavior towards the FSIN property. The set ofpositive integers is denoted by N and U is the Samuel uniformity of the uniformdiscrete space N . The uniformity U is sometimes called the precompact reflectionof the uniform discrete space N . A basis of U is given by the sets ∪ i ≤ n A i × A i ,where A , . . . A n is a partition of the integers. Let N N be endowed with the uni-formity V of uniform convergence when N (the target space) is equipped withthe uniformity U . Let G denote the permutation group of the set N of positiveintegers and let S be the normal subgroup of G given by finitary permutations g ∈ G ; that is, g ∈ S iff the set supp( g ) = { x ∈ N : g ( x ) = x } is finite. It is easy N PROXIMAL FINENESS OF TOPOLOGICAL GROUPS 9 to see that G (hence S ) is a topological group when equipped with the topology τ induced by the uniformity V . More precisely, G is a non-Archimedean group,since a basis of neighborhoods of its unit is given by the subgroups of G of theform H π = { g ∈ G : g ( A i ) = A i , i = 1 , . . . , n } , where π = { A , . . . , A n } is a finite partition of N .The so-called natural Polish topology τ on G , given by pointwise convergence,is coarser than τ and for every partition π of N the set H π is τ -closed. Thisis to say that τ is a cotopology for τ in the sense of [1]; in particular, ( G, τ ) issubmetrizable and Baire. In what follows, unless otherwise stated, the groups G and S will be systematically considered under the topology τ .For later use, we check that the quotient group G/S (with the quotient topol-ogy) is Hausdorff, that is, S is closed in G . Let g ∈ G \ S . A simple inductionallows to construct an infinite set A ⊂ N such that g ( A ) ⊂ N \ A (alternatively, A is obtained from Lemma 2.3 applied to the set { ( x, g ( x )) : x ∈ supp( g ) } ).Let π = { A, N \ A } . Then gH π is a τ -neighborhood of g and for each h ∈ H π , gh ( A ) = g ( A ) ⊂ N \ A . Hence gH π ∩ S = ∅ .Recall that for a topological group H , the lower uniformity U l ∧ U r on H iscalled the Roelcke uniformity and has base consisting of the sets { ( x, y ) ∈ H × H : x ∈ V yV } , V ∈ V ( e ) (see [15]). Every (right) proximally fine group is proximallyfine with respect to the Roelcke uniformity; therefore, the following shows in astrong way that the groups G and S are not proximally fine: Proposition 3.1.
Let k ∈ N and φ : G → N be the evaluation function φ ( g ) = g ( k ) , N being equipped with the discrete uniformity. (1) The function φ : G → N is left uniformly continuous and right proximallycontinuous. In particular, φ is Roelcke-proximally continuous. (2) If N is endowed with the uniformity U , then φ is right uniformly contin-uous. Conversely, if V is uniformity on N such that the restriction of φ | S : S → ( N , V ) is right uniformly continuous, then V ⊂ U .In particular, G and S are not Roelcke proximally fine.Proof.
1) Clearly, φ is left uniformly continuous with respect to the natural topol-ogy τ ; since τ ⊂ τ , φ is left uniformly continuous. To show that φ is right proxi-mally continuous, let L ⊂ G and put π = { A, N \ A } , where A = { g ( k ) : g ∈ L } ,and let us verify that φ ( H π L ) ⊂ φ ( L ). Proposition 2.1 will then conclude the proof. Let h ∈ H π and g ∈ L . Then h ( g ( k )) ∈ { g ( k ) : g ∈ L } , hence we canwrite h ( g ( k )) = g ( k ) for some g ∈ L , thus φ ( h g ) ∈ φ ( L ).2) φ : G → ( N , U ) is right uniformly continuous, since for every partition π = { A , . . . , A n } of N , we have ( g ( k ) , h ( k )) ∈ ∪ i ≤ n A i × A i provided that hg − ∈ H π .For the converse, suppose that V 6⊂ U and let V ∈ V \ U . We will check thatfor any partition π = { A , . . . , A n } of N , there are g, h ∈ S having g ∈ H π h and( φ ( g ) , φ ( h )) V . We may suppose that A = { k } and that A contains twoelements a and b such that ( a, b ) V . Define g ∈ S by g ( k ) = a , g ( a ) = k and g ( x ) = x for x / ∈ { k, a } . Define also h ∈ S by h ( k ) = b , h ( a ) = k , h ( b ) = a and h ( x ) = x otherwise. Then gh − ∈ H π , but ( φ ( g ) , φ ( h )) V . (cid:3) For a subgroup H of G and L ⊂ N , let H ( L ) stand for the pointwise stabilizersof L in H (that is, the set of h ∈ H such that h ( x ) = x for all x ∈ L ). Thefollowing extremal property of τ shows that the above examples are somehowoptimal. Proposition 3.2.
Let H be subgroup of G and F ⊂ N a finite set. Let τ be agroup topology on H and for each k ∈ F , let φ k : g ∈ H → g ( k ) ∈ N , where N isendowed with the discrete uniformity. (1) If for each k ∈ F , φ k is right proximally continuous, then τ is coarserthan τ on H ( N \ HF ) . (2) If for each k ∈ F , φ k is right uniformly continuous, then τ is discrete on H ( N \ HF ) . (3) If τ | H ⊂ τ and ( H, τ ) is strongly FSIN, then τ is discrete on H ( N \ HF ) .Proof.
1) We first show that for a given A ⊂ N , there is a τ -neighborhood V A ofthe unit (in H ) such that f ( A ∩ HF ) ⊂ A for every f ∈ V A . For each k ∈ F ,define L k = { g ∈ H : g ( k ) ∈ A } . According to Proposition 2.1, there is a τ -neighborhood V A of the unit such that for every k ∈ F , φ k ( V A L k ) ⊂ φ k ( L k ). Let n ∈ A ∩ HF and f ∈ V A . Choose g ∈ H and k ∈ F such that g ( k ) = n . Then g ∈ L k , hence φ k ( f g ) ∈ φ k ( L k ) and thus f ( n ) ∈ A . This show that f ( A ∩ HF ) ⊂ A for every f ∈ V A . It follows that for any partition π = { A , . . . , A } of N , there isa τ -neighborhood of the unit in H ( N \ HF ) , namely V = H ( N \ HF ) ∩ V A ∩ . . . ∩ V A n ,such that V ⊂ H π . Since τ is a group topology, it follows that τ is coarser than τ on H ( N \ HF ) .2) Suppose that τ is not discrete on H ( N \ HF ) and let V be τ -neighborhoodof the unit in H . We will show that there are g, h ∈ H and k ∈ F such that N PROXIMAL FINENESS OF TOPOLOGICAL GROUPS 11 gh − ∈ V and g ( k ) = h ( k ), contradicting the right uniform continuity of φ l for atleast one l in the finite set F . Since for each k ∈ F , φ k is τ -continuous, there is f ∈ V ∩ H ( N \ HF ) such that f ( k ) = k for all k ∈ F and f ( a ) = a for some a ∈ N .Then a ∈ HF , hence there are h ∈ H and k ∈ F such that h ( k ) = a . Taking g = f h , we get gh − ∈ V and φ k ( g ) = φ k ( h ).3) If ( H, τ ) is strongly FSIN, then the functions φ k , k ∈ F , are right uniformlycontinuous on H because they are left uniformly continuous on ( G, τ ) and τ | H ⊂ τ . It follows form (2) that τ is discrete on H ( N \ HF ) . (cid:3) Lemma 3.3.
Let H be a subgroup of G , m ∈ N and L ⊂ N such that | L ∩ K | ≤ m for each orbit K of the action of H on N . Then H ( L ) ∈ τ .Proof. There is a finite partition A , . . . , A m of N such that A = N \ L and | A i ∩ K | ≤ ≤ i ≤ m and every orbit K . Then H π ⊂ H ( L ) . Indeed, if f ∈ H π and x ∈ K ∩ A i with i ≥
1, then f ( x ) ∈ K and f ( x ) ∈ A i , thus f ( x ) = x since | A i ∩ K | ≤ (cid:3) Corollary 3.4.
Let H be a subgroup of G for which all but finitely many orbitsare finite and uniformly bounded. Then the discrete topology is the only grouptopology on H that is both proximally fine and finer than τ | H .Proof. If τ is a proximally fine group topology on H finer than τ , then everyfunction φ k is right uniformly continuous (with respect to τ ). It follows fromProposition 3.2(2) that H ( N \ HF ) is τ -discrete for some finite set F ⊂ N such thatthe cardinals of all orbits Hn , n F , are finite and uniformly bounded. ByLemma 3.3, H ( N \ HF ) is τ -open hence τ -open, consequently, τ is discrete. (cid:3) Similarly, the next result follows from Proposition 3.2(3) and Lemma 3.3.
Corollary 3.5.
Let H be subgroup of G for which all but finitely many orbitsare finite and uniformly bounded. If ( H, τ ) strongly FSIN, then ( H, τ ) is discrete.Moreover, if H has finitely many orbits and ( H, τ ) is strongly FSIN, then H isa ( closed ) discrete subgroup of ( G, τ ) . It follows Proposition 3.1 that G and S are not strongly FSIN, hence not SIN,but this does not allows us to conclude that G and S are not FSIN, because noneof the functions φ k , k ∈ N , is bounded. For a topological group H , let R ( H ),respectively U ( H ), stand for the real Banach spaces of bounded right uniformlycontinuous and of bounded right and left uniformly continuous functions on H . Proposition 3.6.
The groups G , S and G/S are not FSIN. Moreover, the densitycharacter of the quotient Banach space R ( G/S ) /U ( G/S ) is at least c .Proof. We shall exhibit, in (1) below, a real-valued bounded function which is leftuniformly continuous on G but not right uniformly continuous when restrictedto S . It will follows that G and S are not FSIN. As for G/S , our strategy is asfollows: For each nonprincipal ultrafilter p on N , we shall give in (2) a boundedright uniformly continuous φ p defined on G which is not left uniformly continuous.This function is in addition constant on every coset of S , so it factorizes to G/S .Then, we show that for each bounded left uniformly continuous function ψ on G ,we have || ψ p + ψ q + ψ || ≥ p , q . Thiswill imply that the Banach space R ( G/S ) /U ( G/S ) contains a uniformly discreteset of the same cardinality as β N \ N , since the quotient map G → G/S is bothleft and right uniformly continuous.(1) Let χ : G → { , } be the function defined by χ ( f ) = 1 if f (1) ≤ f (2) and χ ( f ) = 0 otherwise. Then, clearly, χ is left uniformly continuous. Let us showthat it is not right uniformly continuous on S . Let A , . . . , A n be a partitionof N . We may suppose that A = { a, b } with a = b . Define f and g S by f (1) = g (2) = a , f (2) = g (1) = b , and f ( x ) = g ( x ) for x
6∈ { , } . Then f − ( A i ) = g − ( A i ) for each i = 1 , . . . , n , but χ ( f ) = χ ( g ).(2) Let p be a nontrivial ultrafilter on N and fix an infinite A ⊂ N such that N \ A is infinite. Let ψ p : G → { , } be the function given by ψ p ( f ) = 2 if f − ( A ) ∈ p .Clearly, the function ψ p is bounded and right uniformly continuous. To show that ψ p is constant on every coset of S , let g ∈ G and f ∈ S . Then, for every B ⊂ N , g − ( B ) \ supp( f ) ⊂ ( gf ) − ( B ). Thus, taking B = A if g − ( A ) ∈ p or B = N \ A if not, we get that ψ p ( gf ) = ψ p ( g ).Let p and q be two distinct nonprincipal ultrafilters on N and let us verifythat || ψ p + ψ q + ψ || ≥ ψ : G → R .It will follows that the quotient R ( G/S ) /U ( G/S ) contains norm 1 discrete copyof β N \ N . Let ε > π = { B , . . . , B n } be a partition of N such that | ψ ( f ) − ψ ( g ) | < ε for every f, g ∈ G such that g ∈ f H π . We may suppose that B = C ∪ D with C ∈ p , D ∈ q and C ∩ D = ∅ . Write again D = D ∪ D ,where D and D are infinite, disjoint and D ∈ q . Finally, let { E, F, K } bea partition of of N \ A , with E and F infinite and | K | = | N \ B | . There arecertainly f, g ∈ G such that f ( C ) = A , f ( D ) = E ∪ F , g ( C ) = F , g ( D ) = E , N PROXIMAL FINENESS OF TOPOLOGICAL GROUPS 13 g ( D ) = A and f = g on N \ B . We have f − g ∈ H π (i.e, f ( B i ) = g ( B i ) foreach i ≤ n ), ψ p ( f ) + ψ q ( f ) = 2 and ψ p ( g ) + ψ q ( g ) = 0. Since | ψ ( f ) − ψ ( g ) | < ε ,it follows that | ψ p ( f ) + ψ q ( f ) + ψ ( f ) | ≥ − ε or | ψ p ( g ) + ψ q ( g ) + ψ ( g ) | ≥ − ε .Since ε is arbitrary, we have || ψ p + ψ q + ψ || ≥ (cid:3) The fact that the Hausdorff group
G/S is not discrete was established and usedby Banakh et al. in [4] to answer a question by Dikranjan in [7]. Knowing thatthe symmetric group G and its finitary subgroup S are highly nonabelian (theircenters are trivial) and taking Corollary 3.5 into account, we are naturally led toconclude by asking the following: Question 3.7.
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D´epartement de Math´ematiques, Universit´e de Rouen, UMR CNRS 6085, Av-enue de l’Universit´e, BP.12, F76801 Saint-´Etienne-du-Rouvray, France.
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