Today, I shall use the theorem of Michael discussed earlier to prove that a metric space is paracompact.

Theorem 1 (Stone) A metric space is paracompact.

This theorem seems to use the axiom of choice, or some version thereof, in all proofs.

1. Proof of Stone’s theorem

Suppose given a cover {\mathfrak{A}=\left\{U_\alpha\right\}} of the metric space {X} (with metric {d}, say). We will show that there is a refinement of {\mathfrak{A}} that can be decomposed into a countable collection of locally finite families. Thanks to Michael’s theorem, this will prove the result.