Let $X,Y$ be topological spaces and let $C^0(X,Y)$ be the set of continuous functions between them, endowed with the compact-open topology. I am interested in the following kind of questions:
- What is known in general about the topology of $C^0(X,Y)$? here I am especially interested in path-connectedness, compactness, and simply-connectedness.
- How much does the topology of $C^0(X,Y)$ depend on that of $X$ and/or $Y$? I mean: should I expect $C^0(X,Y)$ to be nicer for a nice choice of $X$ and $Y$? or is there some uniform obstruction/property?
Of course I realize the setting may be too vague, so I should say that I am mostly interested in the case of $X$ and $Y$ manifolds (maybe compact) and in the case $Y=\mathbb{R}$ for a generic $X$. In both cases the compact-open topology coincides with the topology of compact convergence, since the topology on $Y$ is metrizable.
The question I ask here includes some references that you might find interesting (any input on the problem would be appreciated too).
Basically, when $X$ is core-compact there is a topology you can put on $C^{0}(X,Y)$ for any $Y$ (called the Isbell topology) such that it satisfies many nice properties:
When $Y$ is also core-compact, you also get that the composition map is continuous where $C^{0}(Y,Z)$ is given the Isbell topology also (and $Z$ is any topological space).
For spaces $X$ that are locally-compact Hausdorff the compact-open topology on $C^{0}(X,Y)$ is the same as the Isbell topology.
So far the discussion has only depended on $X$. But $Y$ also influences the topology. When $Y$ is $T_0$, $T_1$, Hausdorff, or regular, then so is the Compact-open topology. (I lack a reference to say whether this is true of the Isbell topology).