Pseudocompactness and closed subsets of products

Abstract

Abstract. This paper contains several new characterizations of arbitrary pseudocompact spaces, i.e. spaces characterized by the property that all continuous real-valued functions on the space are bounded. These characterizations parallel known characterizations of Hausdorff spaces including the useful and well-known result that a space Y is Hausdorff if and only if <j> = a whenever <f> and a are continuous functions on a common domain into Y which agree on a dense subset of the domain.

Pseudocompactness via graphs and projections

“…In this article, we give characterizations of pseudocompact spaces which are not necessarily completely regular 1) in terms of graphs of functions into the space, and 2) in terms of projections ; both of these -graphs and projections -are utilized in conjunction with the class M of zero-dimensional metric spaces to effect the characterizations. These characterizations parallel characterizations which have been found to be useful for compact spaces and characterizations which have been given recently for compactness generalizations [9]-[ll], [13]. Pseudocompact spaces which are not necessarily completely regular have been studied in [17].…”