When is the compact-open topology locally compact?

Question

Let $X$ and $Y$ be topological spaces, and consider the compact-open topology on $C(X,Y)$, which is generated by open sets of the form $$\{\text{continuous }f\colon X\to Y:f(K)\subseteq U\}\text{ for compact }K\subseteq X\text{ and open }U\subseteq Y.$$

Without assuming that $Y$ is a metric space, are there conditions on $X$ and $Y$ that imply that the compact open topology on $C(X,Y)$ is locally compact?

One related result of particular importance is the fact that $\mathrm{Hom}(G,T)$ is locally compact, where $G$ is a locally compact Hausdorff abelian group, and $T$ is the circle group. This is important for Pontryagin duality, because it is one part of knowing that the Pontryagin dual of a locally compact Hausdorff abelian group is still a locally compact Hausdorff abelian group.

For this related result, the standard proof uses equicontinuity and Arzela-Ascoli. But this approach only works if $Y$ is a metric space.

Answer 1

This is not a complete answer, but gives some useful information. Let us assume that $Y$ is Hausdorff.

1. Necessary condition: If $C(X,Y)$ is locally compact, then $Y$ is locally compact.
   It is well-known that $j : Y \to C(Y,X), j(y) =$ constant map with value $y$, is an embedding. If $Y$ is Hausdorff, then $j$ is a closed embedding which proves 1.

2. Sufficient condition: If $Y$ is locally compact and $X$ is a finite discrete space, then $C(X,Y)$ is locally compact.
   This is true because $C(X,Y)$ is homeomorphic to a finite product of copies of $Y$.

3. If $Y$ is locally compact non-compact and $X$ is an infinite discrete space, then $C(X,Y)$ is not locally compact.
   This is true because $C(X,Y)$ is homeomorphic to an infinite product of copies of $Y$.

4. If $X$ is an infinite compact Hausdorff space, $C(X,\mathbb R)$ is not locally compact.
   It is well-known that for compact Hausdorff $X$ the compact open topology on $C(X,\mathbb R)$ agrees with the $\sup$-norm topology. Thus $C(X,\mathbb R)$ is a normed linear space which is locally compact if and only if it is finite-dimensional. But for an infinite $X$ the space $C(X,\mathbb R)$ is not finite-dimensional. To see this, let $i : A \hookrightarrow X$ be the inclusion of a closed subspace $A$. It induces a linear map $i^* : C(X,\mathbb R) \to C(A,\mathbb R), i^*(f) = f \mid_A$. By the Tietze extension theorem this map is onto. But for each $n$ the set $X$ contains finite subsets $A_n$ with $n$ elements, thus we get linear surjections $C(X,\mathbb R) \to C(A_n,\mathbb R) \approx \mathbb R^n$.

This shows that it is quite difficult to get a locally compact $C(X,Y)$.

Answer 2

In general $C_K(X,Y)$ (the continuous functions from $X$ to $Y$ in the compact-open topology) will rarely be locally compact (see the other answer and its comments).

But I found in my copy of the Encyclopedia of General Topology (h-1, topological groups, by Dikranjan) facts on the special case of $Y=\Bbb T$ (the circle group ) and its subspace the characters (not all continuous maps). I believe that Abstract Harmonic Analysis (Vol. 1) by Hewitt and Ross has all the proofs. One defines the dual of a group $G$ as:

$$\hat{G} = \{\chi: G \to \Bbb T\mid \chi \text{ a continuous group homomorphism }\} \subseteq C_K(G,\Bbb T)$$

in its subspace topology (so with relativised subbase elements).

• $G$ is compact iff $\hat{G}$ is discrete.
• $G$ is discrete iff $\hat{G}$ is compact.
• If $G$ is LC (locally compact Hausdorff) and Abelian (in total: LCA), then $\hat{G}$ is LC.

The last fact is what you're after. And in that case ($G$ LCA) the Pontryagin-van Kampen duality theorem tells us that $$\omega_G: G \to \hat{\hat {G}}\,;\, \omega_G(g)(\chi) = \chi(g)$$ is a topological group isomorphism.

But that is all quite specific for this subspace and seemingly not a consequence of general facts about the topology of $C_K(X,Y)$ in general. This question also asks about the duality facts. Tao has online notes with the proofs too, IIRC. The question refers to them... These notes also seem useful and do use Arzela-Ascoli for local compactness, so general facts on $C_K(X,Y)$, really.