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 $\omega \times \omega$-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 $\omega$-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.