---- > [!definition] Definition. ([[quotient topology]]) > If $X$ is a [[topological space]] and $A$ is a set and if $p:X \to A$ is a [[surjection]], then there exists exactly one [[topological space|topology]] on $A$ relative to which $p$ is a [[topological quotient map|quotient map]]; it is called the **quotient topology induced by $p$**. > \ > The open sets $U \subset A$ in this topology are those with preimage $p^{-1}(U)$ open in $X$. - [ ] [[TODO]] definition with [[universal property]]/as terminal topology > [!justification] > It is easy to check that this indeed is a topology. $f^{-1}(A)=X$ and $f^{-1}(\emptyset)=\emptyset$ by [[surjection|surjectivity]] of $p$. Since unions and [[preimages and unions commute]], we have $\bigcup_{\alpha}^{} f^{-1}(U_{\alpha})=f^{-1}(\bigcup_{\alpha}^{}U_{\alpha})$ from which stability under arbitrary unions follows. Finite intersection stability follows from the fact that [[preimages and intersections commute]]. ---- #### ---- #### 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 > ```