---- > [!theorem] Theorem. ([[characterization of quotienting a topological space]]) > Let $X,Y$ be [[topological space|topological spaces]] and $\pi:X \to Y$ a [[topological quotient map]]. Let $f:X \to Z$ be any [[continuous|topological map]] that is constant on fibers of $\pi$[^1]. Then $f$ factors uniquely through $Y$ via $\pi$: there exists a unique [[continuous|topological map]] $\tilde{f}$ making the diagram [[universal property|commute]]: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0QBfU9TXfIRRkAjFVqMWbAJrdeIDNjwEiI8uPrNWiEAC1u4mFADm8IqABmAJwgBbJGRA4ISNSAZ0ARjAYAFfspCIFZYxgAWOCDUmlI6ADpxaFhyljb2iI7OSABM0ZLaIBYphWmu1FmIuRJabAkwAB5YcDhwAAQAhK0JeAywwBZcUe5ePv5Kgmwh4ZFcFFxAA > \begin{tikzcd} > X \arrow[d, "\pi"'] \arrow[r, "f"] & Z \\ > Y \arrow[ru, "\exists ! \tilde{f}"'] & > \end{tikzcd} > \end{document} > ``` > > [^1]: For example, if $\pi: X \to X/{\sim}$ is the projection corresponding to an [[equivalence relation]] on $X$, then this condition is stipulating that $f$ is constant on [[equivalence class|equivalence classes]]. > [!proof]- Proof. ([[characterization of quotienting a topological space]]) > The usual argument: if such $\tilde{f}$ exists, then it necessarily is given by $\tilde{f}\big( \pi(x) \big)=f(x)$. Such is well-defined provided $\pi(x)=\pi(y)$ $\implies$ $f(x)=f(y)$, i.e. provided $f$ is constant on $\pi$-fibers. Finally, $\tilde{f}$ is [[continuous]]: let $U \subset Z$ open. Since $\pi$ is a [[topological quotient map|quotient map]], $\tilde{f} ^{-1}(U)$ is open in $Y$ if and only if $\pi ^{-1} \tilde{f} ^{-1} (U)=f^{-1}(U)$ is open in $X$. This is clear since $f$ is [[continuous]]. ---- #### ----- #### 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 > ```