colimit of CW-complexes

Question

For any $n\geq 1$ let $(X_n,X_{n-1})$ be a CW-pair, that is we assume that $X_n$ is a CW-complex and $X_{n-1}$ a subcomplex.

I'm trying to show that $X:=\mathsf{colim} \, X_n$ is a CW-complex. Any help?

Answer 1

The key point about the topology on a CW-complex $X$ is not so much the definition as "closure finite with the weak topology" (Hence CW!), but that continuity of a function $f:X \to Y$ can be decided by seeing if the restrictions of $f$ to all skeletons are continuous; the colimit idea has the same property! Of course to go from $X^n$ to $X^{n+1}$ you need to look at continuity on each $n$-cell.

There is a case for making this the defining property of a CW-complex.