CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
For every topological space $X$ there is a CW complex $Z$ and a weak homotopy equivalence $f :Z\to X$. Such a map $f :Z\to X$ is called a CW approximation to $X$.
This is the notion of cofibrant replacement in the standard model structure on topological spaces.