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There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure. In the Mixed model structure $\mathsf{MixTop}$ ( mixed model structure), weak equivalence are weak homotopy equivalences, fibrations are Hurewicz fibrations and cofibrations are determined by the lifting property.

Suppose that we have a continuous map $f:X\rightarrow Y$ between cofibrant objects in $\mathsf{MixTop}$ and $f$ is a closed embedding i.e. $f(X)$ is closed subspace of $Y$ and $f:X\rightarrow f(X)$ is a homemorphism.

I was wondering if $f$ is a cofibration in $\mathsf{MixTop}$? Here is the nLab reference for the mixed cofibrations.

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No. Let $X$ be an uncountable set and consider $I^X$ with the product topology. Then the inclusion $\{ 0 \} \to I^X$ is a closed embedding between (strongly) contractible spaces which are therefore mixed cofibrant. However, this map is not a Hurewicz cofibration since it would follow that $\{ 0 \}$ is the zero set of a continuous function $I^X \to I$ which would contradict the fact that $0$ has no countable neighbourhood basis in $I^X$. In particular, this map is not a mixed cofibration. (However, it is a Dold cofibration so pushouts along it are still homotopy pushouts with respect to both homotopy equivalences and weak homotopy equivalences.)

Added: Here is an example that is not even a Dold cofibration. Let $Y$ be the subspace of $\mathbb{R}^2$ that is the union of the line segment connecting $(0, 1)$ to $(0, 0)$ and the line segments connecting $(0, 1)$ to $(2^{-m}, 0)$ for all $m \in \mathbb{N}$. The inclusion $\{ 0 \} \to Y$ is a closed embedding, again between strongly contractible spaces. If it was a Dold cofibration, then it would admit a strong deformation retraction (this should be somewhere in the reference I mention in the comment). However, it is a standard exercise in topology that no such retraction exists.

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  • $\begingroup$ Thank you very much, do you have a reference for Dold cofibrations? The map $f$ in the question is not a Dold cofibration ? $\endgroup$ – Ilias Amrani Nov 9 '15 at 12:54
  • $\begingroup$ The only reference I know is Section I.2 of Homotopietheorie by tom Dieck, Kamps and Puppe (in German, they call Dold cofibrations "Homotopiecofaserungen"). I suspect that closed inclusions between mixed cofibrant spaces are not in general Dold cofibrations, but I can't think of a counterexample at the moment. $\endgroup$ – Karol Szumiło Nov 9 '15 at 13:33

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