I would like to know a proof that, given $X$ a topological space, the map $X \to X^I$, $x \mapsto \textrm{const}_x$ is a Hurewicz cofibration (I know how to prove that it is in fact a homotopy equivalence).
I can believe that the image is closed (though a proof or reference would be also nice) and I am allowed to use the retraction argument, but I'm unable to give a reasonable map.