Maximal pseudocompact spaces

Abstract

If P is a topological property and C is a class of topologies, then a space X is said to be maximal P in the class C if X has P but no strictly stronger topology on X which belongs to the class C has P. Recall that a topological space X (with no separation axiom assumed) is feebly compact (called lightly compact in [1]) if every locally finite family of non-empty open subsets of X is finite, or equivalently if every countable nested family of regular closed sets has non-empty intersection. It is well-known that in the class of Tychonoff spaces, feeble compactness is equivalent to pseudocompactness and hence maximal pseudocompactness is equivalent to maximal feeble compactness in the class of Tychonoff spaces.