Take the 2-minute tour ×  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.  share|improve this question add comment  1 Answer
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