The Stacks project

5.4 Separated maps

Just the definition and some simple lemmas.

Definition 5.4.1. A continuous map $f : X \to Y$ of topological spaces is called separated if and only if the diagonal $\Delta : X \to X \times _ Y X$ is a closed map.

Lemma 5.4.2. Let $f : X \to Y$ be continuous map of topological spaces. The following are equivalent:

  1. $f$ is separated,

  2. $\Delta (X) \subset X \times _ Y X$ is a closed subset,

  3. given distinct points $x, x' \in X$ mapping to the same point of $Y$, there exist disjoint open neighbourhoods of $x$ and $x'$.

Proof. Omitted. $\square$

Lemma 5.4.3. Let $f : X \to Y$ be continuous map of topological spaces. If $X$ is Hausdorff, then $f$ is separated.

Proof. Clear from Lemma 5.4.2. $\square$

Lemma 5.4.4. Let $f : X \to Y$ and $Z \to Y$ be continuous maps of topological spaces. If $f$ is separated, then $f' : Z \times _ Y X \to Z$ is separated.

Proof. Follows from characterization (3) of Lemma 5.4.2. $\square$

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