What is the $p$-completion of a space? How is the $p$-completion of a topological space made? I can't find a paper or a book where it is explained, so I will appreciate if anyone answers me with an explanation of how the $p$-completion is done and (optionally) references to read about it.
 A: You have several books at your disposal for this. Possible references are:


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*Bousfield and Kan, Homotopy limits, completions and localizations. /!\ Warning /!\ Contrary to what one might believe, you need to start with the second part of this book to have a chance of understanding the first one!

*May and Ponto, More concise algebraic topology. This is the sequel to May's book A concise course in algebraic topology that Mike Miller mentioned in the comments.


The basic idea of completion is that you want to study your topological space "one prime at a time" basically. Just like you might want to study an abelian group by studying first its 2-torsion, then its 3-torsion... and so on, and then the torsion free part, then you might also want to do the same for a topological space.
If I give you a topological space $X$ then all the information is jumbled together, but if you $p$-complete it then you get a new topological space that only retains the "mod $p$" information of $X$, whatever that might mean -- for example you only keep the mod $p$ (co)homology, but it's more than that. And since there's also a "torsion-free" part, you also want to consider the rationalization of $X$, that only keeps the information over $\mathbb{Q}$. You can now study all these parts independently, and you can mix them back together to have information about $X$.
Note that there is also something called "$p$-localization" of a space. Both topics are generally treated at the same time (both of the books I mentioned do both), the difference with $p$-completion is a difference of finiteness.
