Quotient map

A quotient map is a surjective continuous map $q:X\to Y$ for which $U\subseteq Y$ is open exactly when $q^{-1}(U)$ is open.

If $q:E\to B$ is a quotient map, then $q\times {\mathrm{id}}_{[0,1]}$ is a quotient map. More generally the second factor may be any locally compact space that is Hausdorff. To prove the assertion, let $W\subseteq B\times [0,1]$ have open preimage $O$, and take $(b,t)\in W$. Choose $e\in q^{-1}(b)$ and a compact neighbourhood $K$ of $t$ with $\{e\}\times K\subseteq O$. The set $E_K=\{u:\{u\}\times K\subseteq O\}$ is open by compactness of $K$, and is saturated under $q$ because membership in $O$ depends only on $q(u)$ and the second coordinate. Hence $q(E_K)$ is open by the quotient topology. The rectangle $q(E_K)\times \mathrm{int}K$ is a neighbourhood of $(b,t)$ contained in $W$, proving that $W$ is open. This justifies parameterized homotopies defined on quotient spaces.