D-7 - Pseudocompact Spaces

Abstract

Researchers have found a number of useful conditions equivalent to pseudo compactness. Every countably compact space is pseudo compact. This implication can be divided into a number of sub implications. When combined with any of a variety of other conditions, pseudo compactness implies or nearly implies compactness. Every regular T1-, locally feebly compact space is a Baire space. Every pseudo compact space that is real compact or metrizable is compact. Each regular feebly compact T1-space (for example, every pseudo compact Tychonoff space) that is weakly paracompact is compact. For definitions of these concepts, there are proofs or references to proofs. While countable compactness is inherited by closed subsets, the examples in this chapter illustrate that in general, this is not the case with any of the other conditions considered. If suitable restrictions are satisfied, pseudo compactness is productive. The product of a family of topological groups can be called pseudo compact if and only if (iff) each group in the family is pseudo compact.