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)
54A35 Consistency and independence results, See also {03E35}
54D30 Compactness
03E25 Axiom of choice and related propositions, See also {04A25}
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.
\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.
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.