The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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