Margin between a compact subspace and an including open subspace

In Chapter 31 (“Compact-Open Topology”), the fact that, on the set of the continuous mappings the second topology (the uniform convergence topology) is finer than the first (the compact-open topology) is said to be “clear” but actually needs some non-trivial proof. Part of this is the lemma that I state and prove below.

For any  and real , let  be the open ball centered on and with radius . We know that an open ball is an open subset.

The lemma says:

Let be a metric space, an open subspace of and a compact subspace of . Then there exists such that for all in , .

Proof:

For each , since , there exists an such that .

The collection of all with is an open cover of . Hence there exists a finite such that covers .

Since is finite, there exists a real such that for all , .

Let be an eventual point of .

Since  covers , there exists such that , that is, .

Let be an eventual point of .

We have and , which leads to , that is, . But is in and was taken such that . Hence .

Hence .

We thus have such that .