Remark
Tag 0C1L, permanent. Cite it as [Bourbaki, Tag 0C1L].
General Topology, chapter IX, § 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces
Axiom (O_{IV}) implies (O_{III}) (cf. Chapter I, § 8, no. 4), for if V is a neighbourhood of x_0 and if f is a continuous real-valued function on X with values in [0, 1], such that f(x_0) = 0, f(x) = 1 for all x \in \overline{V}, then the set \overline{f}([0, 1/2]) is a closed neighbourhood of x_0 contained in V. In particular, every completely regular space is regular (which justifies the terminology). But there are examples of regular spaces which are not completely regular (), so that (O{III}) does not imply (O_{IV}).
Every compact space is completely regular (Chapter II, § 4, no. 1, Theorem 1) and therefore so is every subspace of a compact space. We can now complete this proposition by proving its converse:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
Nothing.