Hausdorff compactifications in ZF

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.