Admissible topologies for groups of homeomorphisms and substitutions of groups of \(G\)-spaces
The group of homeomorphisms in the discrete topology acts continuously and transitively on a homogeneous space \(X\). This paper aims at substituting the discrete acting group of a \(G\)-space by the quotient group with respect to the kernel of action, which would preserve various properties (transitivity, being a coset space, preserving a fixed equuniformity in case of \(G\)-Tychonoff space) of actions (Theorem 3.1). Such substitution is interesting from the perspective of investigating metrizability of phase spaces (Theorem 3.27) and deriving bounds on cardinal invariants of acting groups with admissible topologies (Theorem 3.12, 3.13). The authors exploit the product structure of the class of \(\mathcal{G}\)-range topological groups and the presence of a uniform structure on \(G\)-spaces to achieve their goals.
scientific article; zbMATH DE number 7187680