Convergence of sets on a Hausdorff space (looking for a published reference)

Question

I need a reference for the following facts:

• that on $T_1$ spaces, limit and cluster points coincide
• that if $(E,d)$ is a compact metric space, then $(\mathcal{K}(E),d_H)$ is also a compact metric space, where $\mathcal{K}(E)$ is the set of all compact subsets and $d_H$ the Hausdorff metric.

I actually found proofs in a few lecture notes online but I need something published to cite as a reference. Since these are standard, I assume they will be available in most basic books on topology, but I don't have an immediate physical access to many of those. Thanks!

Answer 1

For the second fact you could use the classic paper by E.Michael Topologies on spaces of subsets, which defines the "finite topology" on the hyperspace (aka the Vietoris topology) and shows that this topology coincides with the uniform topology induced by the Hausdorff metric, and then he shows that for Vietoris we have that the hyperspace of $X$ is compact $T_2$ iff $X$ is. This is where I learnt this from.