Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$.
Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular simplex. There's a natural map $f:|Sing(X)|\to X$ and there's a theorem that says that $f$ is a weak homotopy equivalence. That is, $f$ induces isomorphisms of homotopy groups.
Then it seems that Whitehead theorem applies and gives that $f$ is homotopy equivalence as long as $X$ is homotopy equivalent to a CW complex (i.e. $X$ is m-cofibrant). Is that correct?
Is there an example when $f$ is not homotopy equivalence? Any examples that come up in "real life"?