----- > [!proposition] Proposition. ([[Hausdorff quotient need not be Hausdorff]]) > Let $\sim$ be the [[equivalence relation]] on $\mathbb{R}$ s.t. $x \sim y$ if and only if $x-y$ is [[rational]]. Then $\mathbb{R} / \sim$ with the [[quotient topology]] is not [[Hausdorff space|Hausdorff]]. > [!proof]- Proof. ([[Hausdorff quotient need not be Hausdorff]]) > > Recall that the difference of any two rational numbers is rational. Hence all rational numbers are equivalent to all other rational numbers. Every irrational number $i$ is equivalent to each other irrational number $j$ having the property $j=i+q$ for some rational $q$. > > Thus, the [[equivalence class|equivalence classes]] of $\sim$ are all of $\mathbb{Q}$ and $\{ i \}$ for all $i \in \mathbb{R} \to \mathbb{Q}$. > > An open [[neighborhood]] of a point in $\mathbb{R} /\sim$ is a collection of [[equivalence class|equivalence classes]] whose union is open in $\mathbb{R}$ and contains the point. The rationals are not open in $\mathbb{R}$. Neither is any subset of the irrationals. Neither is any set obtained by adjoining a subset of the irrationals to the rationals, unless the union equals $\mathbb{R}$. > > Therefore, the only open neighborhood of a point in $\mathbb{R} / \sim$ turns out to be $\mathbb{R}$ itself! This obviously implies that $\mathbb{R} / \sim$ cannot be Hausdorff. > ----- #### ---- #### 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 > ```