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Recall that we call a map $i: A \rightarrow X$ a cofibration if it has the homotopy extension property. We will say a pointed space $X$ is well-pointed, if the inclusion of the basepoint $\{ * \} \hookrightarrow X$ is a cofibration.

A pointed cofibration $i: A \rightarrow X$ is a based map of pointed spaces that has the homotopy extension property with respect to homotopies respecting the basepoint. Note that a cofibration is always a pointed cofibration, but the converse is not true.

It is stated in May's "Concise Course in Algebraic Topology", although not proved, that if a map of well-pointed spaces is a pointed cofibartion, then it is already a cofibration. I've been trying to do it on my own using "the box method", but it didn't lead me anywhere.

How does one go to prove such statements? Is there any general method, any useful tricks?

I assume that all spaces are compactly generated weakly Hausdorff.

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This is lemma 1.3.4 in May's "More concise algebraic topology", it looks very technical.

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