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.