If a $k$-space is a weak P-space, is it discrete?

Question

Let $Y$ be a locally compact Hausdorff space and $q:Y\to X$ a quotient map with $X$ Hausdorff (so $X$ is a $k$-space). Suppose that $X$ is a weak P-space (i.e. every countable subset of $X$ is closed). Is $X$ discrete?

If $Y$ is also assumed to be $\sigma$-compact then the answer is yes, because then $X$ is $\sigma$-compact and every compact subspace of a weak P-space is finite. What about the general case?

Answer 1

So you know that every compact subspace of $X$ is finite. This gives in combination with being a $k$-space that $X$ is discrete:

Let $O \subseteq X$ be any subset of $X$. If $K$ is any compact subspace of $X$, it's obvious that $O \cap K$ is open in $K$ (as $K$ is finite and thus discrete), and as $K$ was arbitrary and $X$ is a $k$-space, $O$ is open in $X$. So $X$ is discrete.