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In Paul Selick's book Introduction to Homotopy Theory, he says that one can prove the CW-Approximation Theorem:

Given a topological space $Y$ there exists CW-complex $X$ and a map $f : X \rightarrow Y$ such that $f_\ast : \pi_n(X) \rightarrow \pi_n(Y)$ is an isomorphism for all $n$.

using the fact that

Let $f : X \rightarrow Y$ be a map between CW-complex. Then there exists $g : X \rightarrow Y$ such that $g \simeq f$ and $g$ is cellular.

However he does not give any detail and I don't see how one should proceed.

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  • $\begingroup$ Paul Selick's book is very concise. It it not really a good "introduction". I would recommend considering another book for a learner. For example, Hatcher's text, which covers CW approximation on pg 352. (Personally I dislike Hatcher's style, but his book still comes highly recommended.) $\endgroup$ – Tyrone May 30 '18 at 9:31
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In Hatcher in his book available here : http://www.math.cornell.edu/~hatcher/AT/AT.pdf describes "cellular approximation" that results in theorem 4.8 page 349, which is what you want to prove.

As usual with CW-complexes, this is prove by induction on the $n$-dimensional skeletons. The details can be quite complex depending on what you already know true about CW-complexes or not, I'm letting you read the text and ask another question if you don't understand.

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  • $\begingroup$ Thanks for you answer. I know the proof of Hatcher's book. What I am really wondering is how to get the result from this proposition, as a "second proof" if you wish. $\endgroup$ – Oscar P. May 30 '18 at 10:05

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