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I know the following result:

If $X$ is a compact smooth manifold and $G$ is a compact Lie group which acts smoothly on $X$, then $X_G = (X\times EG)/G$ is a CW complex.

I don't know how to prove this result. Where can I find the proof? Thank you!

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I think you can just take $EG$ to be a CW complex, and $G$ will act freely on the product? –  Aaron Mazel-Gee Apr 15 '11 at 6:30
In this proposition, smooth manifold and smooth action may be necessary in some sense. Because when X is a local finite CW complex, X$\times$EG is a CW complex but the assertion is generally wrong. –  Boh Liu Apr 15 '11 at 7:00

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