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](http://ncatlab.org/nlab/show/model+structure+on+topological+spaces)), 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](http://ncatlab.org/nlab/show/m-cofibrant+space) is the nLab reference for the mixed cofibrations. 



Different model structures on Top
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.


