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?
$C_k(X,\mathbb R)$ is Baire;
$C_k(X,\{0,1\})$ is Baire.
I suspect that such questions should be considered in the literature. Could you give me any references?
