---- > [!definition] Definition. ([[normal topological space]]) > Let $X$ be a [[topological space]] in which singletons are [[closed set|closed sets]]. > $X$ is called **normal** if for each pair $A,B$ of disjoint [[closed set|closed sets]] of $X$, there exists disjoint open sets containing $A$ and $B$, respectively. > ![[CleanShot 2024-04-20 at 11.19.36@2x 1.jpg]] > [!equivalence] > ![[characterization of normal spaces#^7b099d]] ^028382 > [!basicproperties] > - Normal spaces are regular, and [[regular topological space|regular spaces]] are [[Hausdorff space|Hausdorff]]. (Note that we need closed singletons for this to be the case; a two-point space with the [[discrete topology|indiscrete topology]] satisfies the other parts of the normality/regularity conditions, even though it is not [[Hausdorff space|Hausdorff]].) > >- The normality axiom is *stronger* than the regularity axiom (and in turn the Hausdorff axiom). [[TODO]] examples from Munkres that show this > [!basicexample] > - [[regular and second-countable implies normal]] > - [[metric spaces are normal]] > - [[compact and Hausdorff implies normal]] ---- #### ---- #### 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