On a Notion of Weak Compactness in Non-Regular Spaces

Abstract

This chapter discusses a notion of weak compactness that can be considered as either a strengthening of absolute closure or a weakening of relative compactness depending upon whether one considers entire topological spaces or whether one restricts to subspaces of larger spaces. While these notions are reduced to standard compactness notions in regular spaces, the existence of large families is noted of nonregular Hausdorff spaces that have these new properties but are far from compact. Nevertheless, these properties are not too different from their standard counterparts in that, for example, they are preserved under products and images of continuous functions. The chapter also discusses the analogs of some known generalizations of sequential compactness and shows that various productivity theorems of Scarborough and Stone, Saks and Stephenson, and Frolik, Booth, and the author hold for these new notions. These theorems are then used to construct spaces not having the standard properties but rather only the weak properties.