Universality on spaces continuously containing topological groups

Abstract

In 1986 V.V. Uspenskij proved that there exists a universal topolog-
ical group with a countable base and in 1990 put the problem: does there exist a
universal topological group of weight an uncountable cardinal τ. 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 τ for the collection of all topological groups of weight τ. In the
present paper we prove that in the class of all topological spaces of weight τ,
which are continuously containing spaces for a collection of topological groups,
there are universal elements.