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This is a question I asked at Math.SE but got no answers:

Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes singular cohomology, if $X$ is a finite CW complex.

Somewhere on the Internet I saw the statement $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \check{\mathrm{H}} {}^\ast(X; \mathbb{Q})$, where $\check{\mathrm{H}} {}^\ast$ denotes Cech cohomology, if $X$ is a compact Hausdorff space.

First, I'm looking for a reference for this fact $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \check{\mathrm{H}} {}^\ast(X; \mathbb{Q})$.

Second, can the statements be extended to non-compact spaces, i.e., do we have something like $\mathrm{K}^\ast_{\text{cpt}}(X) \otimes \mathbb{Q} \cong \mathrm{H}_{\text{cpt}}^\ast(X; \mathbb{Q})$ for (locally finite) CW complexes? Or something analogous for the Cech cohomology and locally compact Hausdorff spaces?

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For a compact Hausdorff space $X$ you can describe both Cech cohomology and K-theory as direct limits over finite complexes that $X$ maps to. For locally compact Hausdorff spaces you might want to consider the one-pojnt compactification. – Tom Goodwillie May 21 '13 at 14:08
up vote 1 down vote accepted

I have found a reference:

"M. Karoubi, Les isomorphismes de Chern et de Thom-Gysin en K-theorie, Seminaire Henri Cartan, vol. 16, no. 2, Expose no. 16, 1963-1964".

There the isomorphism $\mathrm{K}^0(X) \otimes \mathbb{Q} \cong \mathrm{\check{H}}^{ev}(X; \mathbb{Q})$ is shown for every compact Hausdorff space $X$. But the Theorem 2 from there should also be applicable to $\mathrm{K}^\ast(X) \otimes \mathbb{Q}$ and $\check{\mathrm{H}}^\ast(X; \mathbb{Q})$.

For locally compact Hausdorff spaces $X$ one can indeed just consider the one-point compactification $X^+$ and apply the above isomorphism to the pair $(X^+, \infty)$.

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