(Co-)fibrations in Top and CGWH Suppose that you have a map $i: A\rightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a cofibration in Top (the category of topological spaces)? What about Hurewicz fibrations?

A map $i: A\to X$ is a cofibration in Top iff the canonical map $l: M_i\to X\times I$ has a retract iff $Z=X\times \\{0\\}\cup i(A)\times I$ is a retract of $X\times I$. Knowing that a map is cofibration in CGWH is enough to conlude that $l$ has a retract, because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH.

Concerning the last question, the answer is negative, see the comments.