Universality on spaces continuously containing topological groups

Abstract

In 1986 V.V. Uspenskij proved that there exists a universal topological group with a countable base and in 1990 put the problem: does there exist a universal topological group of weight an uncountable cardinal ?? This problem is still open. In 2015 we gave the notion of a continuously containing space for a given collection of topological groups and proved that there exists such a space of weight ? for the collection of all topological groups of weight ? ?. In the present paper we prove that in the class of all topological spaces of weight ? ?, which are continuously containing spaces for a collection of topological groups, there are universal elements.