Proposition 2

Tag 0B68, permanent. Cite it as [Bourbaki, Tag 0B68].

General Topology, chapter I, § 8. Hausdorff spaces and regular spaces

Let $f, g$ be two continuous mappings of a topological space $X$ into a Hausdorff space $Y$ ; then the set of all $x \in X$ such that $f(x) = g(x)$ is closed in $X$ .

For this set is the inverse image of the diagonal of $Y \times Y$ under the mapping $x \mapsto (f(x), g(x))$ , which is continuous (§ 4, no. 1, Proposition 1). The result therefore follows from (Hii) and § 2, no. 1, Theorem 1.

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

Cited by

Proposition 5, § 10
Proposition 8, § 11
Proposition 10, § 8
Corollary 2, § 8
Theorem 1, § 8
Theorem 3, § 3
Proposition 2, § 2
Proposition 3, § 2
Proposition 6, § 3
Remark 2, § 6
Proposition 9, § 6
§ 4. Exponentials and logarithms
§ 1. Subgroups and quotient groups of $\mathbf{R}^n$
Exercise 7, § 1
Proposition 7, § 1
Proposition 4, § 3

Cites

§ 2, no. 1, Theorem 1
§ 4, no. 1, Proposition 1