On subcompactness and countable subcompactness of metrizable spaces in ZF

TL;DR

This paper investigates the relationships between subcompactness and countable subcompactness in metrizable spaces within ZF set theory, establishing equivalences and independence results for these properties.

Contribution

It proves key equivalences between subcompactness notions and classical compactness in metrizable spaces in ZF, and shows some implications are independent of ZF.

Findings

Every subcompact metrizable space is completely metrizable.

A metrizable space is countably compact iff it is countably subcompact.

The negation of certain implications is relatively consistent with ZF.

Abstract

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space X=(X,T) is countably compact iff it is countably subcompact relative to T. (iii) For every metric space X=(X,d), X is compact iff it is subcompact relative to T. We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable space is completely metrizable, (b) every countably subcompact metrizable space is subcompact, (c) every complete metrizable space is subcompact is relatively consistent with ZF.