----- > [!proposition] Proposition. ([[characterization of normal spaces]]) > Let $X$ be a [[topological space]]. Let singletons in $X$ be [[closed set|closed]]. Then $X$ is [[normal topological space|normal]] if and only if given closed $A \subset X$ and an open set $U \supset A$, there is an open set $V \supset A$ s.t. $\overline{V} \subset U$. ^7b099d > [!proof]- Proof. ([[characterization of normal spaces]]) > Same argument from [[characterization of regular spaces]] goes through if you replace the point $x$ by the set $A$ throughout. ----- #### ---- #### 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 > ``` #reformatrevisebatch04