An Observation on Initially $\kappa $-Compact Spaces

TL;DR

This paper extends a known compactness result from countably compact spaces to initially κ-compact spaces with a quasi G_κ-diagonal, showing such spaces are compact.

Contribution

It generalizes Chaber's theorem to initial κ-compactness, broadening the class of spaces where quasi G_κ-diagonals imply compactness.

Findings

Initially κ-compact spaces with a quasi G_κ-diagonal are compact.

The result applies to any infinite cardinal κ.

Generalizes previous countable compactness results.

Abstract

In \cite{Chaber}, Chaber has proved that countably compact spaces with a quasi $G_{\delta }$-diagonal are compact. We prove that initially $\kappa $% -compact spaces with a quasi $G_{\kappa }$-diagonal are compact, for any infinite cardinal $\kappa .