The collection of all compact finite subsets is $F_{\sigma}$ in the hyperspace of all compact subsets.

Question

Let $X$ be metrizable (not necessarily Polish), and consider the hyperspace of all compact subsets of $X$, $K(X)$, endowed with the Vietoris topology (subbasic opens: $\{K∈K(X):K\subseteq U \}$ and $\{K\in K(X):K\cap U\neq \emptyset\}$ for $U\subseteq X$ open), or equivalently, the Hausdorff metric. We want to show that $$K_{f}(X)=\{K\in K(X):K \text{is finite}\}$$ is $F_{\sigma}$ in $K(X)$.

I've been thinking about this exercise and I have not been able to solve.

Any ideas?

Answer 1

HINT: For $n\in\Bbb N$ let $\mathscr{K}_n(X)=\{K\in \mathscr{K}(X):|K|\le n\}$, and show that each $\mathscr{K}_n(X)$ is closed.