----- > [!proposition] Proposition. ([[Hausdorff iff diagonal is closed]]) > A [[topological space]] $X$ is [[Hausdorff space|Hausdorff]] if and only if the **diagonal** $\Delta :=\{ (x,x) : x \in X \}$ > is closed in $X \times X$ with respect to the [[product topology]]. > [!proof]- Proof. ([[Hausdorff iff diagonal is closed]]) > $\to$. Suppose that $X$ is [[Hausdorff space|Hausdorff]]; we'll show this implies $(X \times X) \cut \Delta$ is [[closed set|open in]] $X$. This will be accomplished by showing that $(X \times X) \cut \Delta$ [[open sets can be nestled into|can be nestled into]]. So, let $(a,b) \in X \times X$ and let $U \times V \in X \times X$ be a [[basis for a topology|basis element]] containing $(a,b)$. Using that $X$ is [[Hausdorff space|Hausdorff]], obtain [[neighborhood]]s $A \ni a$ and $B \ni b$ with $A \cap B = \emptyset$. Then $(A \cap U) \times (B \cap V)$ is an [[open set]] in $(X \times X)$ containing no elements of the form $(x,x)$, and is thus contained in $(X \times X) \cut \Delta$. $\textcolor{Apricot}{\text{why not just use A and B directly and skip U and V?}}$ > \ > $\leftarrow$. Suppose $\Delta$ is not [[Hausdorff space|Hausdorff]]. Then there exist $x_{1},x_{2} \in X$ such that for all [[neighborhood]]s $U_{1} \ni x_{1}$, $U_{2} \ni x_{2}$, we have $U_{1} \cap U_{2} \neq \emptyset$. This implies that $(U_{1} \times U_{2}) \cap \Delta \neq \emptyset$ for all $U_{1} \ni x_{1}$ and $U_{2} \ni x_{2}$. In turn, we must have that $(X \times X) \cut \Delta$ is not [[open set|open in]] $X \times X$ by [[open sets can be nestled into]]: we've found a point $(x_{1},x_{2}) \in (X \times X) \cut \Delta$ about which no [[neighborhood]] $U_{(x_{1},x_{2})}$ exists with $(x_{1},x_{2}) \in U_{(x_{1},x_{2})} \subset (X \times X) \cut \Delta$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```