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?

  • Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X \times I$. Using Strickland 1.10 we see that if a map $i : A \to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG? – Paul Frost Dec 6 at 16:20
  • @PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you – Gregg Dec 6 at 18:02
  • Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577) – Gregg Dec 6 at 18:06
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    If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X \times 0 \cup i(A) \times I$ is a retract of $X \times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X \times I$: $Z$ is closed in $X \times I$, hence a CGWH space. – Paul Frost Dec 6 at 19:54
  • @PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_i\to X\times I$ has a retract iff $Z$ is a retract of $X\times I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed 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). Thank you! – Gregg Dec 8 at 19:40
up vote 1 down vote accepted

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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