Closed subsets of compact-like topological spaces

Abstract

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of ω×ω-matrix units cannot be embedded into a topological semigroup which is a weakly H-closed topological space. We show that each Hausdorff topological space is a closed subspace of some ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. Also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a positive solution of a problem posed by Banakh, Dimitrova, and Gutik.

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

“…Compact topological semigroups cannot contain an isomorphic copy of the bicyclic monoid [2]. The problem of an embedding of the bicyclic monoid into compact-like topological semigroups was discussed in [4,5,12,25,26]. However, it is natural to consider the bicyclic monoid with an adjoint zero C 0 .…”

A countably compact topological group with the non-countably pracompact square

“…The productivity of other compact-like properties can be a non-trivial problem. For instance, see [19], [13], [5], [12], [2], and [7]. In [16], [17], and [18] Tkachenko considered the productivity of various properties of topological groups.…”