A generalization of $\varkappa$-metrizable spaces

TL;DR

This paper introduces countably $ varkappa$-metrizable spaces, expanding the class of $ varkappa$-metrizable spaces and exploring their properties, especially in relation to pseudocompact spaces and compactifications.

Contribution

It defines countably $ varkappa$-metrizable spaces, shows their relation to $ varkappa$-metrizable spaces, and generalizes a key result about their compactifications.

Findings

Countably $ varkappa$-metrizable spaces form a broader class than $ varkappa$-metrizable spaces.

For pseudocompact spaces, the classes coincide.

The eche9-Stone compactification of such spaces remains $ varkappa$-metrizable.

Abstract

We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with $\varkappa$-metrizable spaces. We prove a generalization of a Chigogidze result that the \v{C}ech-Stone compactification of a pseudocompact countably $\varkappa$-metrizable space is $\varkappa$-metrizable.