Hausdorff compactifications in ZF

TL;DR

This paper explores properties of Hausdorff compactifications within ZF set theory, revealing that certain classical theorems fail without the axiom of choice and establishing conditions for generating compactifications.

Contribution

It demonstrates the existence of non-equivalent compactifications with identical function algebras in ZF, and provides conditions for generating compactifications without relying on the axiom of choice.

Findings

Existence of non-equivalent compactifications with equal function algebras in ZF

Failure of Glicksberg's theorem in some ZF models

Conditions for generating compactifications from function sets

Abstract

For a compactification $\alpha X$ of a Tychonoff space $X$, the algebra of all functions $f\in C(X)$ that are continuously extendable over $% \alpha X$ is denoted by $C_{\alpha}(X)$. It is shown that, in a model of $\textbf{ZF}$, it may happen that a discrete space $X$ can have non-equivalent Hausdorff compactifications $\alpha X$ and $\gamma X$ such that $% C_{\alpha}(X)=C_{\gamma}(X)$. Amorphous sets are applied to a proof that Glicksberg's theorem that $\beta X\times \beta Y$ is the Cech-Stone compactification of $X\times Y$ when $X\times Y$ is a Tychonoff pseudocompact space is false in some models of $\mathbf{ZF}$. It is noticed that if all Tychonoff compactifications of locally compact spaces had $C^{\ast}$-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space $X$ to generate a compactification of $X$ are given in $\mathbf{ZF}$. A concept of a functional \v{C}ech-Stone compactification is investigated in the absence of the axiom of choice.