Properties:: [[every finite point set in a Hausdorff space is closed]], [[limits in Hausdorff spaces are unique]], [[Hausdorff product is Hausdorff]], [[Hausdorff subspace is Hausdorff]], [[Hausdorff quotient need not be Hausdorff]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: [[Hausdorff iff diagonal is closed]] ---- > [!definition] Definition. ([[Hausdorff space]]) > A [[topological space]] $X$ is called a **Hausdorff space** if for each pair $x_{1},x_{2}$ of distinct points of $X$, there exist [[neighborhood]]s $U_{1}$ and $U_{2}$ of $x_{1}$ and $x_{2}$, respectively, that are disjoint. > ![[CleanShot 2024-04-20 at 11.19.36@2x 2.jpg]] ---- #### ---- #### 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 > ```