Gδ-open functionally bounded subsets in topological groups

Abstract

This paper deals with functionally bounded sets of a topological group which are G δ -dense in their closure in the bilateral completion of the group. We prove that the functionally bounded sets with this property coincide, for topological groups, with a kind of functionally bounded sets introduced by Isiwata: the hyperbounded sets. We prove that when B is a hyperbounded set of a topological group G which has a G δ -open subset dense in B, then B has only one compatible uniformity, the one inherited from G. We also prove that for any functionally bounded G δ -open subset A of a topological group G, the space cl G A belongs to the Frolik's class B , i.e., the Cartesian product of cl G A by any pseudocompact space is a pseudocompact space. As a consequence we get that every compact topological group is the Stone-Čech compactification of any of its G δ -dense subsets.