**MSC:**- 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

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: