I am looking for models of ${\mathsf{Top}}$ distinct from modifications of simplicial sets. ~~The above question should be understandable to the reader. I'll add more details when I get access to a proper computer.~~ Let ${\mathsf{Met}}$ be the category of metric spaces and continuous maps. Then there is an embedding ${\mathsf{Met}}\hookrightarrow {\mathsf{Top}}$. Is this embedding a Quillen equivalence?

**Edit:** Professor May explains below that my question is not precise, per se, as there is more than one Quillen inequivalent model categorial structures on ${\mathsf{Top}}$.

**Edit:** The answer is no. Tom commented that the category ${\mathsf{Met}}$ does not have all small colimits. Thus the embedding ${\mathsf{Met}}\hookrightarrow {\mathsf{Top}}$ cannot be a Quillen equivalence.

uniformlycontinuous maps has many finite colimits - not all, but enough for the purposes of Baues' cofibration category (or Brown's category of cofibrant objects), arxiv.org/abs/1106.3249 – Sergey Melikhov Feb 6 '12 at 16:25