  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.
