Corollary 1 (Principle of extension of identities)

Tag 0B69, permanent. Cite it as [Bourbaki, Tag 0B69].

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 $Y$ . If $f(x) = g(x)$ at all points of a dense subset of $X$ , then $f = g$ .

In other words, a continuous map of $X$ into $Y$ (Hausdorff) is uniquely determined by its values at all points of a dense subset of $X$ .

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

Cited by

Proposition 8, § 11
Theorem 1, § 8
Corollary, § 3
Theorem 3, § 3
Proposition 2, § 2
Proposition 5, § 3
Proposition 6, § 3
Theorem 1, § 6
§ 4. Exponentials and logarithms
Proposition 7, § 1

Cites

Nothing.