I have a technical question. My terminology:
I - set of standard inclusions $\partial I^n \to I^n$.
I-cell (Relative Cell Complexes) - transfinite compositions of pushouts of maps in $I$.
CW (CW complexes) - the usual definition (like I-cell but with "cells attached by order of dimension).
I-cof - retracts of maps in I-cell, the same as maps having Left Lifting Property w.r.t. maps having Right lifting property w.r.t. I (by Quillen small object argument).
My first question is to confirm that CW in I-cell in I-cof are all different
My second question is: In this page: http://ncatlab.org/nlab/show/model+structure+on+topological+spaces it is written under "Mixed model structure" (the model structure on $Top$ for which equivalences are weak equivalences and fibrations are Hurewicz fibrations) that cofibrant objects are spaces homotopy equivalent to CW complexes. Is this exactly true (or is it for example spaces homotopy equivalent to cell complexes?). I also read somewhere around nlab that Milnor advocated that spaces homotopy equivalent to CW complexes are nice, so I wanted to know if the cofibrant objects in this mixed model structure are exactly such, or a bit more general.
Thank you, Sasha