nLab
CW approximation

Idea

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.

References

• Allen Hatcher, Algebraic topology, Cambridge Univ. Press 2002; Chapter 4, Section 4.1 pdf