Horst Herrlich
The Ascoli Theorem is equivalent to the Boolean Prime Ideal Theorem
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 51, 137-140(1997)
MSC:
54D30 Compactness
03E30 Axiomatics of classical set theory and its fragments
Abstract: It is well-known that in \textbf{ZF} (i.e., Zermelo-Fraenkel set
theory without the Axiom of Choice) the following hold:
\begin{Theo-o}\hspace{-1.5ex}\textbf{\large{\em\cite{her3}}} The
Tychonoff Product Theorem is equivalent to the Axiom of Choice.
\end{Theo-o}
\begin{Theo-o}\hspace{-1.5ex}\textbf{\large{\em\cite{her5}}} The
\v{C}ech-Stone Theorem is equivalent to the Boolean Prime Ideal Theorem.
\end{Theo-o}
What is the corresponding status of the Ascoli Theorem\,? It is the purpose
of this note to settle this question. Since the Ascoli Theorem occurs in a
variety of forms (see the comprehensive study in \cite{her4}), the form
used here needs to be specified (although the title-result is rather
stable). For the purpose of this paper the following version is used:
\begin{Theo-Ascoli}If $\mathbf X$ is a locally compact Hausdorff space,
$\mathbf Y$ is a metric space, $C_{co}(\mathbf X,\mathbf Y)$ is the space
of all continuous maps from $\mathbf X$ to $\mathbf Y$ with the
compact-open topology, and $F$ is a subspace of $C_{co}(\mathbf X,\mathbf Y)$, then the following conditions are equivalent:
Notes: Abstract contains the first few lines of text of the paper.