$\mathbf{E}$-compact extensions in the absence of the Axiom of Choice

TL;DR

This paper investigates fundamental properties of E-compact extensions, Hewitt realcompactifications, and Banaschewski compactifications of spaces without relying on the Axiom of Choice, revealing new results and equivalences.

Contribution

It presents new results on E-compact extensions and related compactifications in ZF set theory, including properties of specific subrings and topologies, some of which are equivalent to choice principles.

Findings

New results on E-compact extensions without the Axiom of Choice

Characterizations of Hewitt realcompactifications and Banaschewski compactifications in ZF

Connections between topological properties and choice principles

Abstract

The main aim of this work is to show, in the absence of the Axiom of Choice, fundamental results on $\mathbf{E}$-compact extensions of $\mathbf{E}$-completely regular spaces, in particular, on Hewitt realcompactifications and Banaschewski compactifications. Some original results concern a special subring of the ring of all continuous real functions on a given zero-dimensional $T_1$-space. New facts about $P$-spaces, Baire topologies and $G_{\delta}$-topologies are also shown. Not all statements investigated here have proofs in $\mathbf{ZF}$. Some statements are shown equivalent to the Boolean Prime ideal Theorem, some are consequences of the Axiom of Countable Multiple Choices.