Characterizing topological spaces $X,Y$ whose function space $C_k(X,Y)$ is Baire

Question

I am looking for a characterization of topological spaces $X,Y$ for which the function space $C_k(X,Y)$ is Baire. Here $C_k(X,Y)$ is the space of continuous functions from $X$ to $Y$, endowed with the compact-open topology.

It is well-known that for any complete metric space $Y$ and and compact space $X$ the space $C_k(X,Y)$ is metrizable by the complete metric $d(f,g):=\max_{x\in X}d_Y(f(x),g(x))$.

This implies that for a $k_\omega$-space $X$ and any Polish space $Y$ the function space $C_k(X,Y)$ is Cech-complete and hence Baire.

On the other hand, for any $k$-space $X$ and any Polish space $Y$ the function space $C_k(X,Y)$ is Dieudonne-complete (being homeomorphic to a closed subspace of the Tychonoff product $\prod_{K\in\mathcal K(X)}C_k(K,Y)$ of Cech-complete spaces, where $\mathcal K(X)$ is the family of all compact subsets of $X$).

If $X$ a sequential space, then $C_k(X,Y)$ is Hewitt-complete (being homeomorphic to a closed subspace of the Tychonoff product $\prod_{K\in\mathcal{MK}(X)}C_k(K,Y)$ of Polish spaces, where where $\mathcal{MK}(X)$ is the family of all compact metrizable subsets of $X$).

Nonetheless Hewitt-complete or Dieudonne-complete spaces need not be Baire.

Problem 1. Is there a $k$-space $X$ whose function space $C_k(X,\mathbb R)$ or $C_k(X,\{0,1\})$ is Baire but not Cech-complete?

Problem 2. Let $X$ be a zero-dimensional $k$-space (containing a unique non-isolated point). Are the following two statements equivalent?

1. $C_k(X,\mathbb R)$ is Baire;

2. $C_k(X,\{0,1\})$ is Baire.

I suspect that such questions should be considered in the literature. Could you give me any references?

Answer 1

The Baireness of $C_k(X)=C_k(X,\mathbb{R})$ has been characterized for some subclasses of $k$-spaces. For example, Gruenhage and Ma proved that for a locally compact or first-countable space $X$, the following are equivalent:

1. $C_k(X)$ is Baire.
2. $X$ has the Moving Off Property.

Moving Off Property means that if $\mathcal{K}$ is a collection of compact subsets of $X$ such that every compact subset of $X$ is disjoint from some member of $\mathcal{K}$ (such a collection is called "Moving Off"), then $\mathcal{K}$ contains an infinite subcollection with a discrete open expansion.

(1) --> (2) is easily seen to be true, even without the assumption that $X$ is locally compact or first-countable (hint: if $\mathcal{K}$ is a moving off collection then $O_n=\{f \in C_k(X): (\exists K \in \mathcal{K}) (f(K)>n)\}$ is a dense open subset of $C_k(X))$), but it is an open problem whether the above characterization holds for every (completely regular) space $X$.

In the last section of their paper, Gruenhage and Ma offer an example of a locally compact space $X$ such that $C_k(X)$ is Baire but not weakly $\alpha$-favorable. Since every Cech-complete space is weakly $\alpha$-favorable, their example solves your Problem 1.