---- > [!definition] Definition. ([[topological quotient map]]) > Let $X$ and $Y$ be [[topological space|topological spaces]]; let $p:X \to Y$ be a [[surjection]]. The map $p$ is said to be a **quotient map** provided a subset $U$ of $Y$ is open in $Y$ if and only if $p ^{-1}(U)$ is open in $X$. > \ > This is a stronger notion than [[continuous|continuity]]. It includes the [[continuous]] [[open map|open maps]] (and [[continuous]] [[closed set|closed maps]]) as special cases. > [!equivalence] > To say that $p$ is a quotient map is equivalent to saying that $p$ is [[continuous]] and $p$ maps [[saturated set|saturated]] open sets of $X$ to open sets of $Y$. ^equivalence > [!basicexample] > Let $X$ be the [[subspace topology|subspace]] $[0,1] \cup [2,3]$ of $\mathbb{R}$, and let $Y$ be the [[subspace topology|subspace]] $[0,2]$ of $\mathbb{R}$. The map $p:X \to Y$ defined by $p(x) = \begin{cases} x & x \in [0,1] \\ x-1 & x \in [2,3] \end{cases}$ is [[continuous]], surjective, and [[closed set|closed]]; therefore it is a quotient map. It is not an [[open map]], though; the image of the [[open set]] $[0,1]$ of $X$ is not open in $Y$. --- #### ---- #### 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 > ```