Is CW approximation interesting?

Question

Each topological space $X$ is weakly equivalent to a CW-complex $X'$. That is, there exits a map $f:X'\rightarrow X$ that induces isomorphisms on $\pi_n$ for each $n\in \mathbb N$.

My question is about how deep is this theorem. How can we use it ? and in which context can we replace a topological space by its weakly homotopy equivalent CW-approximation especially that the theorem does not give any information about the correponding CW-space $X'$, no topological information neither combinatorial such as its CW-decompostion! Thank you for your help!

Answer 1

The point is simply that, if you want to prove some property P for arbitrary topological spaces (which can be nasty to work with), and you happen to know that P is preserved by weak homotopy equivalence, it will often suffice to prove P only for CW-complexes (which have a nice, concrete, combinatorial description).

Properties preserved by weak homotopy equivalence include all homotopical and homological data about your spaces. So, for instance, in order to prove the Künneth formula for general spaces, it suffices to prove it for CW-complexes only.

If you prefer, you can interpret the role of this theorem in the context of algebraic topology as a whole, simply by asking yourself "why do I care about this?". Here are some example ways you might read it:

• Why are we able to study arbitrary spaces at all, when they're mostly very hard? Because CW-approximation allows us to pass interesting information from a nice, much simpler class of objects.
• Why do we mostly study homotopical data? Because it's the most accessible sort of data to calculate, because of results like CW-approximation.
• Why do we spend so much time studying CW-complexes? Because they're a near-optimal balance between simple things (objects we can do calculations with) and interesting things (objects we actually care about), precisely because of the existence of a CW-approximation theorem.

Ultimately, it doesn't matter how you view it; the effect is much the same. But I've often found putting it in context helps you see the bigger picture. The above are all grossly oversimplified viewpoints, but they're probably all at least part of the truth, and perhaps one of them is a way of thinking about it that you hadn't encountered before.