Proposition 3
Tag 0C1M, permanent. Cite it as [Bourbaki, Tag 0C1M].
General Topology, chapter IX, § 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces
A topological space X is completely regular if and only if it is homeomorphic to a subspace of a compact space.
Consider the coarsest uniformity on X with respect to which all continuous mappings of X into [0, 1] are uniformly continuous; we have already used this uniformity in the proof of Theorem 2, where we saw that it is compatible with the topology of X if X is uniformizable. Furthermore, this uniformity is a structure of a precompact space, by the compactness of the interval [0, 1] and Proposition 3 of Chapter II, § 4, no. 2. If X is Hausdorff, the completion of X with respect to this is therefore compact, and the proposition is proved.
We can therefore say that a completely regular space can be embedded in a compact space. It is often convenient to present this result in the following way:
In general, a cube is a topological space K^I, the product of a family of topological spaces each identical with a compact interval K of \mathbf{R}, indexed by an arbitrary set I. If I is finite and has n elements, we recover the notion of an n-dimensional closed cube, which was defined in Chapter VI, § 1, no. 1. A cube is a compact space (Chapter I, § 9, no. 5, Theorem 3).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.