On $\kappa$-pseudocompactess and uniform homeomorphisms of function spaces

TL;DR

This paper introduces the concept of $$-pseudocompactness, a generalization of pseudocompactness, and proves that this property can be characterized by the uniform structure of the corresponding function space, answering a question posed by Arhangel'skii.

Contribution

It establishes that $$-pseudocompactness is determined by the uniform structure of the function space, extending known results from pseudocompactness.

Findings

$$-pseudocompactness generalizes pseudocompactness.

The uniform structure of $C_p(X)$ characterizes $$-pseudocompactness.

Affirmative answer to Arhangel'skii's question about the characterization of $$-pseudocompactness.

Abstract

A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of $X$ into $\mathbb{R}^\kappa$ 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 $C_p(X)$ of continuous real-valued functions on $X$ endowed with the pointwise topology. In respect of that A.V. Arhangel'skii asked in [Topology Appl., 89 (1998)] if analogous assertion is true for $\kappa$-pseudocompactness. We provide an affirmative answer to this question.