TITLE: When is the diagonal inclusion a Hurewicz cofibration
QUESTION [10 upvotes]: Given a space $X$, what conditions on $X$ can you give to ensure that the diagonal map $X\to X\times X$ is a Hurewicz cofibration?  (I am happy to assume that $X$ is compactly generated weak Hausdorff, or even just Hausdorf.)
More generally, given a map $f:X\to Y$, when is $X\to X\times_YX$ a Hurewicz cofibration?

REPLY [8 votes]: The diagonal condition was used crucially in Milnor's classical paper.
Milnor, John. On spaces having the homotopy type of a CW-complex. 
Trans. Amer. Math. Soc. 90 1959 272–280.
He gives earlier references to Fox and Serre.  In the parametrized
generalization, there is a general fiberwise NDR pair characterization
of cofibrations that applies (e.g. Lemma 5.2.4 in May and Sigurdsson,
Parametrized homotopy theory).