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][2], 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 $CW$-complexs, while the cofibrant objects in the Quillen model structure are the topological spaces that are retracts of $CW$-complexs. 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][1] 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 $Top$ in which the map $(0,0)\to I$ is a cofibration.

So let us assume that we are given a standard model structure on $Top$, such that both $(0,0)\to I$ and $I\to W$ are cofibrations. We can consider the over model category structure on $Top_*:=Top_{*/}$. Let $(Top_*)_\infty$ denote the underlying $\infty$ category of this model structure (which is just the well known $\infty$-category of pointed spaces). Let $Ho(Top_*)=Ho((Top_*)_\infty)$ be the associated homotopy category and let
$$H^n:Ho(Top_*)\to Ab$$ denote the usual cohomology functors: $$H^n(X)=Ho(Top_*)(X,K(Z,n)).$$
Let us choose $(0,0)$ as a basepoint for both $I$ and $W$, so that $I\to W$ is a cofibration in $Top_*$. Note that $W/I\cong S^1$ is just the usual circle, so since $I$ is cofibrant and $I\to W$ is a cofibration in $Top_*$, we know that $I\to W\to S^1$ is a cofiber sequence in the $\infty$-category $Top_\infty$ (or in other words $W\to S^1$ is the cofiber of $I\to W$ in the $\infty$-category $Top_\infty$). Thus we have a long exact sequence in cohomology, and, since $I\simeq *$ in $Top_*$, we get that $W$ and $S^1$ have the same cohomology. This contradicts the fact that $W$ is weakly contractible.

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.

I am not sure what this says about David's comment on $\Delta$-generated spaces. Perhaps it means that the Warsaw circle is not $\Delta$-generated? Note, however, that the Warsaw circle is a sequential space, being metrizable, so it is $\mathbb{N}_+$-generated, where $\mathbb{N}_+$ is the one point compactification of the natural numbers. [1]:What are your favorite instructional counterexamples? [2]:http://ncatlab.org/nlab/show/model+structure+on+topological+spaces#BergerMoerdijk03