On $$\kappa $$-Pseudocompactess and Uniform Homeomorphisms of Function Spaces

Abstract

A Tychonoff space X is called $$\kappa $$ κ -pseudocompact if for every continuous mapping f of X into $$\mathbb {R}^\kappa $$ R κ the image f(X) is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying between pseudocompact and compact spaces. It is well known that pseudocompactness of X is determined by the uniform structure of the function space $$C_p(X)$$ C p ( X ) of continuous real-valued functions on X endowed with the pointwise topology. In respect of that A.V. Arhangel’skii asked if analogous assertion is true for $$\kappa $$ κ -pseudocompactness. We provide an affirmative answer to this question.