I have strong intuition that the following fact is true:
If $X = \bigcup_{n\in \mathbb N} X_n$ is a CW-complex (and $X_n$ its $n$-skeleton) then $$ \tilde H_n (X) = \tilde H_n(X_{n+1}). $$ ($\tilde H_*$ denote the reduced homology groups).
I am able to show it for finite dimensionnal CW-complexes (by induction on the degree and using the long exact sequence in homology), but I'm stuck for the infinite dimensionnal case.
Do you have any advice on how to proceed, and if possible using only the Eilenberg–Steenrod axioms?
Thanks.