On κ -Pseudocompactess and Uniform Homeomorphisms of Function Spaces

A Tychonoff space X is called κ-pseudocompact if for every continuous mapping f of X into Rκ 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 Cp(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 κ-pseudocompactness. We provide an affirmative answer to this question.