Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ are defined. One is the Quillen model structure, in which fibrations are the Serre fibrations, and one is the Mixed model structure, in which the fibrations are the Hurewicz fibrations. Since every Hurewicz fibrations is a Serre fibration, The Mixed Model Structure has more cofibrations then the Quillen Model Structure. For e.g., the cofibrant objects in the Mixed model structure are the topological spaces that are homotopy equivalent to $CW$-complexes, while the cofibrant objects in the Quillen model structure are the topological spaces that are retracts of $CW$-complexes. My Question is the following:
Is there a standard model structure on $Top$ with more cofibrations then the Mixed model structure?
Edit: Here is a counterexample that can give a certain upper bound on the class of cofibrations in any standard model structure on $Top$.
Let $W$ denote the Warsaw circle. The Warsaw circle can be defined as the subspace of the plane $R^2$ consisting of the graph of $y = \sin(1/x)$, for $x\in(0,1]$, the segment $[−1,1]$ in the $y$ axis, and an arc connecting $(1,\sin(1))$ and $(0,0)$ (which is otherwise disjoint from the graph and the segment). It can be shown that $W$ is weakly contractible but not contractible (see here for more details).
Let $I$ denote the segment $[−1,1]$ in the $y$ axis. I claim that the embedding $I\to W$ cannot be a cofibration in any standard model structure on $Top$. Indeed, suppose that we are given a standard model structure on $Top$ such that $I\to W$ is a cofibration. Then $I\to W$ is an acyclic cofibration, since $W$ is weakly contractible. Now take the pushout of $I\to W$ along $I\to *$. Since acyclic cofibrations are closed under pushouts, and since $W/I\cong S^1$ is just the usual circle, we get that $*\to S^1$ is an acyclic cofibration. Contradiction.
Note that the Warsaw circle is a compact metrizable space (being a bounded closed subspace of $R^2$) and $I\to W$ is a closed embedding. In particular, we get that there is no standard model structure on $Top$ in which every closed embedding between compact metrizable spaces is a cofibration.