Pseudocompactness properties

Abstract

A topological extension property is a class of Tychonoff spaces P \mathcal {P} which is closed hereditary, closed under formation of topological products and contains all compact spaces. If X X is Tychonoff and P \mathcal {P} is an extension property, there is a space P X \mathcal {P}X such that X ⊆ P X ⊆ β X , P X ∈ P X \subseteq \mathcal {P}X \subseteq \beta X,\;\mathcal {P}X \in \mathcal {P} and if f : X → Y f:X \to Y where Y ∈ P Y \in \mathcal {P} then f f admits a continuous extension to P X \mathcal {P}X . A space X X is called P \mathcal {P} -pseudocompact if P X = β X \mathcal {P}X = \beta X . In this note it is shown that if P \mathcal {P} is an extension property which contains the real line (e.g., the class of realcompact spaces), X X is P \mathcal {P} -pseudocompact and Y Y is compact, then X × Y X \times Y is P \mathcal {P} -pseudocompact. An example is given of an extension property P \mathcal {P} , a P \mathcal {P} -pseudocompact space X X and a compact space Y Y such that X × Y X \times Y is not P \mathcal {P} -pseudocompact.