----- > [!proposition] Proposition. ([[characterization of regular spaces]]) > Let $X$ be a [[topological space]]. Let singletons in $X$ be [[closed set|closed]]. Then $X$ is [[regular topological space|regular]] if and only if given $x \in X$ and an (open) [[neighborhood]] $U \ni x$, there is a [[neighborhood]] $V \ni x$ s.t. $\overline{V} \subset U$. ^001b95 > [!proof]- Proof. ([[characterization of regular spaces]]) > $\to$. Suppose the specified condition holds. Let $x \in X$ and let $B$ be a [[closed set]] in $X$ disjoint from $x$. Then $U:=X-B$ is an open [[neighborhood]] of $x$; fix a [[neighborhood]] $V \ni x$ s.t. $\overline{V} \subset U$. Then $X-\overline{V}$ is a open set containing $B$ which is disjoint from $U$. > $\leftarrow$. Suppose $X$ is [[regular topological space|regular]]. Let $x \in X$ and $U$ an open [[neighborhood]] of $x$. Let $B=X-U$, then $B$ is a [[closed set]]. > Using regularity fix open disjoint sets $V_{1} \ni x$ and $V_{2} \supset B$. The set $\overline{V_{1}}$ is disjoint from $B$, since if $y \in B$, the set $V_{2}$ is a [[neighborhood]] of $y$ disjoint from $V_{1}$. Therefore $\overline{V} \subset U$. ----- #### ---- #### 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