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Let $X$ be a topological space and let $X_\mathbb Q$ be its rationalization.

1) What is the rationalization of $X_\mathbb Q$, is it $X_\mathbb Q$ itself?

2) If $X$ is a CW-complex, does that imply necessarily that $X_\mathbb Q$ is also a CW-complex?

3) If $X_\mathbb Q$ and $X'_\mathbb Q$ are two rationalizations of $X$, how do they relate? My guess is that they are weakly equivalent, and so if they are CW-complexes, then they are homotopy equivalent.

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    Given that rationalization is defined by a universal property, certainly $(X_\mathbb{Q})_\mathbb{Q}\cong X_{\mathbb{Q}}$.2011-05-10

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