Stone's theorem on paracompactness of metric spaces

Remarks

The theorem is due to A. H. Stone. A short proof by Mary Ellen Rudin well-orders the members of the cover. For each $n$ and each member $U$, it takes the union of balls of radius $2^{-n}$ around suitable points whose first containing member is $U$. The radii are chosen so that each point has a ball meeting only finitely many of the new sets. The proof uses the axiom of choice through the well-ordering.

Example at work: every metric space is therefore normal and admits partitions of unity subordinate to any open cover.

The metric is essential only through the topology it induces. So every metrizable space is paracompact.

Going deeper

For a student

Stone's theorem shows that paracompactness is a necessary condition for metrizability. Metrization theorems such as the Nagata–Smirnov theorem go in the other direction. Combined with Dieudonné's theorem, it gives another proof that metric spaces are normal. It is also why paracompactness is automatic for metrizable manifolds.