Localization of a space

Summary

Localizing a space at a set of primes $P$ is the topological analogue of localizing the integers at $P$. For a simply connected space $X$ it produces a map $X \to X_P$. This map tensors every homotopy group and every homology group with $\Z_{(P)}$. This lets one study a space one prime at a time. The empty set of primes gives rationalization, the starting point of rational homotopy theory.