---- > [!definition] Definition. ([[quotient space]]) > Let $X$ be a [[topological space]], and let $X^{*}=\{ [x] \}_{x \in X}$ be a [[partition]] of $X$ into disjoint subsets whose union is $X$ (a [[quotient set]]). Let $p:X \to X^{*}$ be the natural projection map carrying $x$ to $[x]$. In the [[quotient topology]] induced by $p$, the space $X^{*}$ is called the **quotient space** of $X$. > \ > Since any partition uniquely determines an [[equivalence relation]], on can think of $X^{*}$ as having been obtained by 'identifying' each pair of equivalent points in $X$ — "in $X^{*}$, equivalence becomes equality". > \ > The typical open set of $X^{*}$ is a collection of [[equivalence class|equivalence classes]] whose union is an open set of $X$. - [ ] in terms of [[saturated set|saturated sets]] and the like - [ ] define in terms of [[universal property]] of the [[quotient topology]] ---- #### ---- #### 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 > ```