On coverings in the lattice of all group topologies of arbitrary Abelian groups

Abstract

The remainder of the completion of a topological abelian group (G, τ 0 ) contains a nonzero element of prime order if and only if G admits a Hausdorff group topology τ 1 that precedes the given topology and is such that (G, τ 0 ) has no base of closed zero neighborhoods in (G, τ 1 ). This article is a continuation of the study of the lattice of topologies on algebraic systems which was started in [1][2][3][4]. Here we study the properties of the pairs of group topologies on an abelian group which admit no other group topologies between the members of a pair. We propose a method (see Theorem 9) for obtaining the group topologies that precede a given Hausdorff group topology and whose initial topology has no base of closed zero neighborhoods. As demonstrated in Theorem 10, this method yields every such topology.We use the following notations. Denote by N and Z the sets of naturals and integers respectively. If k ∈ Z then |k| is the modulus of k. Denote by G an abelian group in additive presentation. If k ∈ Z and U ⊆ G then putIf M ⊆ G and τ is a topology on G then we denote by [M ] (G,τ ) the closure of M in (G, τ ) and by τ | M , the restriction of τ to M . If τ 0 and τ 1 are group topologies on G such that τ 1 < τ 0 and there are no other group topologies on G between them then τ 0 is called the covering of τ 1 , and we write τ 1 ≺ τ 0 . The expression τ 1 τ 0 means that τ 1 ≺ τ 0 or τ 1 = τ 0 .Remark 1. If A is a subgroup of an abelian group G and τ is a group topology on A then all zero neighborhoods in (A, τ ), considered as subsets in G, constitute a set that satisfies conditions BN1-BN6 (see [5, Theorem 1.2.5]) and, hence, the latter can be taken as a base of zero neighborhoods of some group topology on G. We denote this group topology on G also by τ , which will not lead to confusion, since either the group will be clear from the context on which this topology is considered or the choice of the group is immaterial.Remark 2. Induction on n shows easily that if S 0 , S 1 , . . . is a sequence of subsets in an abelian group G such that S k+1 + S k+1 ⊆ S k for every k ∈ N then t+n i=n S i + S t+n ⊆ S n−1 for all t, n ∈ N. Theorem 3. Suppose that τ 0 and τ 1 are first-countable group topologies on an abelian group G such that τ 1 < τ 0 and (G, τ 0 ) has a base of closed zero neighborhoods in (G, τ 1 ). Then G admits a firstcountable group topology τ such that τ 1 < τ < τ 0 .Proof. Let B 0 and B 1 be countable bases of symmetric zero neighborhoods in (G, τ 0 ) and (G, τ 1 ) respectively such that every zero neighborhood V ∈ B 0 is a zero neighborhood in (G, τ 1 ) but is a closed set in (G, τ 1 ).Chişinȃu (Moldova).

M‐normal subgroups and non‐abelian lower continuous topological groups

“…In this case, τ is called a successor of σ and σ a predecessor of τ. Some authors also prefer to say that τ is a covering of σ ([1, 2]). A Hausdorff topological group G is called lower continuous if its topology does not admit a Hausdorff predecessor.…”