Remark

Tag 0C2G, permanent. Cite it as [Bourbaki, Tag 0C2G].

General Topology, chapter IX, § 2. Metric spaces and metrizable spaces

If $d(x, A) = a$ , it can happen that there is no point of A whose distance from x is equal to a. However, this situation can never arise if A is compact, for then Weierstrass's theorem (Chapter IV, § 6, no. 1, Theorem 1) shows that there exists $y \in A$ such that $d(x, A) = d(x, y)$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Chapter IV, § 6, no. 1, Theorem 1