As nicely expounded at the n-lab, there are three well-known model structures on Top: the Quillen model structure, the Hurewicz model structure, and the mixed model structure, which takes its weak equivalences from the Quillen one and its fibrations from the Hurewicz.
Presumably if we try to mix the model strutures the other way round, it doesn't work. That is, if we try to define a model structure on Top by taking
$\mathcal{W}$ := homotopy equivalences,
$\mathcal{F}$ := Serre fibrations, i.e. maps with the right lifting property w.r.t. $X \rightarrow X \times [0,1]$ for every cell complex $X$ (equivalently, for every disc).
$\mathcal{C}$ := maps with the left lifting prop w.r.t. the inclusion $\mathcal{W} \cap \mathcal{F}$,
then something will go wrong. My suspicion is that it's impossible to construct the "cofibration; trivial fibration" factorisation:
the path-object construction used for this in the Hurewicz structure doesn't work, as the left map may no longer be a cofibration if the spaces involved aren't cell complexes;
the small object argument used in the Quillen structure doesn't work, since as we're now using strong homotopy equivalences not weak ones, there's no longer a generating set of cofibrations.
I haven't however either proved come up with an argument that no such factorisation exists or found a reference for it yet..for... but presumably this too is well-known in the right circles, just less widely written-about than the related positive results?
(Thanks to Michael Warren for pointing me towards this example.)
