= Product of a quotient map with the unit interval
{title2=$q\times\operatorname{id}_{[0,1]}$}
If $q:E\to B$ is a <quotient map>, then $q\times\operatorname{id}_{[0,1]}$ 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 $q(E_K)\times\operatorname{int}K$ is a <neighbourhood> of $(b,t)$ contained in $W$, proving that $W$ is open. This justifies parameterized <homotopies> defined on <quotient spaces>.
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