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?
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$\begingroup$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.
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