The Stacks project

Graphs of maps to Hausdorff spaces are closed.

Lemma 5.3.2. Let $f : X \to Y$ be a continuous map of topological spaces. If $Y$ is Hausdorff, then the graph of $f$ is closed in $X \times Y$.

Proof. The graph is the inverse image of the diagonal under the map $X \times Y \to Y \times Y$. Thus the lemma follows from Lemma 5.3.1. $\square$

Comments (1)

Comment #1226 by David Corwin on

Suggested slogan: Graphs of maps to Hausdorff spaces are closed

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View Lemma 5.3.2 as pdf