The local properties in topological groups and related concepts and questions

Abstract

The next general question is considered below: when certain open subspaces of topological groups are homeomorphic to some topological group? A version of this question has been posed in [4] (open problem 1.4.1). In it, the complements to the elements in any zero-dimensional topological group were considered. A result below in this direction is Theorem 3.1 saying that if G is a σ-compact topological group and F is a nonempty compact subspace of G such that the complement G∖F is homeomorphic to some topological group, then F is a Gδ-subset of G (and therefore, by a remarkable theorem of M.M. Choban [6], F is a dyadic compactum). Another conclusion in this theorem says that if, moreover, the tightness of F is countable (in particular, if F is first countable), or if F is topologically linearly orderable, then G has a countable network. Lemma 4.1 says that if X is a Lindelöf homogeneous space with countable Souslin number and X is puncturable at some point a, then a is a Gδ-point in X, the subspace Y=X∖{a} is Lindelöf, and X is submetrizable. It follows from this that every Lindelöf topological field with countable Souslin number is submetrizable (its topology contains a metrizable topology). A natural Abelian topological group with few open subspaces that are homeomorphic to topological groups is identified in Example 3.2.