Topology and its Applications 160 (2013) 1083–1087
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Topology and its Applications www.elsevier.com/locate/topol
Thin subsets of topological groups I.V. Protasov Department of Cybernetics, Kyiv National University, Ukraine
a r t i c l e
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Article history: Received 12 November 2012 Received in revised form 18 February 2013 Accepted 20 February 2013 MSC: 22A05
a b s t r a c t A subset A of a group G with the identity e is called thin if g A ∩ A and Ag ∩ A are finite for each g ∈ G \ {e }. We prove that each countable totally bounded topological group has a dense thin subset and a thin subset X such that e is the unique limit point of X. On the other hand, for each thin subset T of a countable Abelian group G, we construct a non-discrete group topology on G in which T is closed and discrete. © 2013 Elsevier B.V. All rights reserved.
Keywords: Topological group Thin subset T -sequence
1. Introduction For a group G we denote by PG and FG the families of all and all finite subsets of G. A subset A of a group G with the identity e is said to be
• • • •
large if there exists F ∈ FG such that G = F A = A F ; small if G \ F A F is large for each F ∈ FG ; thin if the sets g A ∩ A and Ag ∩ A are finite for each g ∈ G \ {e }; k-thin for k ∈ N if | g A ∩ A | k and | Ag ∩ A | k for each g ∈ G \ {e }.
All these definitions have left and right versions. A subset A is left (right) large if there exists F ∈ FG such that G = F A (G = A F ). A subset A is left (right) small if G \ F A (G \ A F ) is left (right) large for each F ∈ FG . By [16, Theorem 12.4], A is small if and only if A is left and right small. We note that A is thin if and only if, for each F ∈ FG with e ∈ F , there exists K ∈ FG such that F a ∩ A = {a} = A ∩ aF for each a ∈ A \ K . From this observations, we conclude that each thin subset of an infinite group is small. Every infinite group G can be generated by some small subset [13]. This is an answer to a question of Malykhin and Moresko [11]. Moreover [9], there are 2-thin subset X and 4-thin subset Y such that G = X X −1 ∪ X −1 X = Y Y −1 . Each infinite group G can be partitioned in ℵ0 small subsets [13], but G can be partitioned in ℵ0 thin subsets if and only if |G | ℵ1 [14]. Every thin subset of an infinite amenable group G is a null set with respect to any left or right invariant Banach measure on G, but this does not hold for small subsets [10]. It is worth to be mentioned that large, small and thin subsets can be defined in much more general context of booleans, the asymptotic counterparts of uniform topological spaces [15–17], and can be considered as counterparts of dense, nowhere dense and discrete subsets. In what follows all group topologies are supposed to be Hausdorff, “countable” means “countably infinite”.
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I.V. Protasov / Topology and its Applications 160 (2013) 1083–1087
If an n∈ω is an injective convergent sequence in a topological group G then the subset {an : n ∈ ω} is relatively discrete and thin. Following [5], for an Abelian group G we denote by G # the group G endowed with the strongest totally bounded group topology. Since G # has no injective convergent sequences, van Douwen asked [5, Question 4.17]: if G is infinite, does G # have a relatively discrete subset that is not closed? A positive answer to this question was obtained in [12]: if a topological group G contains an infinite totally bounded subset then there exists a non-closed relatively discrete subset −1 x : m < n < ω} for an appropriate injective sequence x of G. This subset X was constructed in the form X = {xm n n n ∈ω . −1 −1 − 1 Since (xk xm )(xm xn ) = (xk xn ), X is not thin. By [8, Theorem 1.4], every infinite Abelian group G contains a subset A such that | A | = |G | and 0 is the unique limit point of A − A in G # , see also [7] for the case of Boolean groups. We prove (Theorem 2) that each countable totally bounded topological group has a thin subset X such that e is the unique limit point of X . This paper is motivated also by the following result of Bella and Malykhin [1]: each countable Abelian totally bounded group contains a small dense subset. We prove that each countable totally bounded group has a dense thin subset (Theorem 3). We show (Theorem 1) that an infinite group G has a 2-thin dense subset provided that w(G ) d(G ), where w(G ) is a weight of G, d(G ) is a dispersion character of G, the minimal cardinality of non-empty open subsets of G. In particular, this is true for each compact group, but there are non-discrete locally compact groups without dense thin subsets. On the other hand (Theorem 4), for each thin subset T of a countable Abelian group G, we construct a non-discrete group topology on G in which T is discrete and closed. The paper is concluded with some open questions. 2. Groups with w( G ) d( G ) The following lemma is due to C. Chou [2, Proposition 4.1]. Lemma 1. Let G be a group and let X = {xα :
α < κ } be a subset of G, X α = {xγ : γ < α }. If
xα ∈ / X α X α−1 X α for each α < κ then X is 2-thin. Theorem 1. Let G be an infinite topological group. Assume that there exists a family F of subsets of G such that the following conditions are satisfied: (i) |F | |G |, | F | |F | for each F ∈ F ; (ii) for every open subset U of G, there is F ∈ F such that F ⊆ U . Then G has a 2-thin dense subset X such that | X | = |F |. Proof. We enumerate F = { F α : α < |F |}, take an arbitrary element x0 ∈ F 0 and suppose that, for some α < |F |, the / X α X α−1 X α where X α = elements {xγ : γ < α } have been chosen. Since | F α | |F |, we can choose xα ∈ Fα such that xα ∈ {xγ : γ < α }. After |F | steps, we put X = {xα : α < |F |}. By Lemma 1, X is 2-thin. By (i) and the construction of X , X is dense. 2 If w(G ) d(G ), to apply Theorem 1, we take an arbitrary base F of the topology such that |F | = w(G ). Example 1. Let G 1 be non-discrete topological group, G 2 be a discrete group such that |G 2 | > 2|G 1 | . We show that G = G 1 × G 2 has no dense thin subsets. Let X be a dense subset of G. For each g ∈ G 2 , let X g = X ∩ (G 1 × { g }). Since X is dense and G 2 is discrete, the subset X g is infinite for each g ∈ G 2 . We write X g = Y g × { g } and pick distinct g 1 , g 2 ∈ G 2 such that Y g1 = Y g2 . Since X g1 = X g2 ( g 2−1 g 1 )
and X g2 = ( g 2 g 1−1 ) X g1 , we conclude that X is neither right nor left thin.
By Theorem 1, each infinite compact group G has a dense thin subset. Each finite group G has such a subset but by the trivial reason: G is thin. If in Example 1 G 1 is compact, we get a non-discrete locally compact group G without dense thin subsets. For countable non-discrete groups with no dense thin subsets see Question 3 in the last section. 3. Totally bounded groups 1 For a finite ordered subset X = {x1 , . . . , xn } of a group G, we denote D X = {x− i x j : 1 i < j n}. Recall that a topological group G is totally bounded if, for each neighbourhood U of e, there is a finite subset F such that G = F U .
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Lemma 2. Let G be an infinite totally bounded topological group and let A be a subset of G. Assume that, for each n ∈ N, there exists an ordered subset X n = {x1 , . . . , xn } of G such that D X n ⊂ A. Then e is a limit point of A. Proof. We fix an arbitrary neighbourhood U of the identity e, and choose a symmetric neighbourhood V of e such that V 4 ⊆ U . Then we pick F ∈ FG such that G = F V . For n > | F |, by the assumption, there is an ordered subset X n = {x1 , . . . , xn } such that D X n ⊂ A. We choose g ∈ F and distinct i , j, i < j such that xi V ∩ g V = ∅ and x j V ∩ g V = ∅. Since g ∈ xi V V ∩ 1 x j V V , we have x− x j ∈ V V V V ⊆ U . Hence A ∩ U = ∅. i
2
Given F ∈ FG , we say that a subset A of a group G is F -separated if y ∈ / F xF for all distinct x, y ∈ A. We observe that if, for any F ∈ FG , there is K ∈ FG such that A \ K is F -separated then A is thin. Lemma 3. Let G be a non-discrete topological group, F ∈ FG , e ∈ F , F = F −1 , g ∈ G, U be an open neighbourhood of e. For each n > 0, there exist distinct elements x1 , . . . , xn of G such that D {x1 , . . . , xn } ⊂ U and g D {x1 , . . . , xn } is F -separated. Proof. We put x1 = e and suppose that, for some m < n, the elements x1 , . . . , xm have been chosen so that D {x1 , . . . , xm } ⊂ U , 1 −1 {x− 1 , . . . , xm } ⊂ U and the set
1 −1 g D {x1 , . . . , xm } ∪ x− 1 , . . . , xm
1 −1 is F -separated. We choose a symmetric neighbourhood V of e such that {x− 1 , . . . , xm } V ⊂ U ,
1 1 gx− V ∩ F gx− V F = ∅, i j
1 gx− xj ∈ / F gxk−1 V F i
−1 1 −1 for all i , j , k ∈ {1, . . . , m}, i < j. We pick xm+1 ∈ V \ {x1 , . . . , xm } such that gxm / F g {x− 1 , . . . , xm } F . Then D {x1 , . . . , +1 ∈ −1 −1 xm+1 } ⊂ U , {x1 , . . . , xm+1 } ⊂ U and the set
1 −1 g D {x1 , . . . , xm+1 } ∪ x− 1 , . . . , xm+1
is F -separated. After n steps we get a desired set {x1 , . . . , xn }.
2
Theorem 2. Every countable totally bounded topological group G contains a thin subset X such that e is the unique limit point of X .
Proof. We write G as a union G = n∈ω F n of an increasing chain of finite symmetric subsets, e ∈ F 0 . We shall construct a sequence X n n∈ω in FG and a decreasing sequence U n n∈ω of closed neighbourhoods of e inductively. Put X 0 = ∅ and U 0 = G. Assume that, for some n ∈ ω , the subsets X 0 , . . . , X n and neighbourhoods U 0 . . . U n have been chosen so that, for each i ∈ {0, . . . , n}, | X i | = i, (1) D X i ⊂ U i −1 \ U i ; (2) D X i is F i -separated; (3) F i ( D X i ) F i ∩ ( D X 0 ∪ · · · ∪ D X i −1 ) = ∅. Then we choose U n+1 such that U n+1 ⊂ U n , ( D X 0 ∪ · · · ∪ D X n ) ∩ U n+1 = ∅, and apply Lemma 3 to find X n+1 so that | Xn+1 | = n + 1, D Xn+1 ⊂ U n+1 , D X n+1 is F n+1 -separated and F n+1 ( D X n+1 ) F n+1 ∩ ( D X 0 ∪ · · · ∪ D X n ) = ∅. limit point of X . If we choose U n n∈ω so that n∈ω U n = {e } After ω steps, we put X = n∈ω D X n . By Lemma 2, e is a then, by (1), e is the unique limit point of X . By (2) and (3), i n D X n is F n -separated. Hence X is thin. 2 Theorem 3. Each countable totally bounded topological group G has a thin dense subset.
Proof. Let G = { gn : n ∈ ω}, and G is written as a union G = n∈ω F n of an increasing chain of finite symmetric subsets, e ∈ F0. In view of Lemma it suffices to find, for each i ∈ ω , a sequence X i j i j
