This is a question I asked at Math.SE but got no answers: <https://math.stackexchange.com/q/397164/7110/>


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.



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> First, I'm looking for a reference for this fact $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \check{\mathrm{H}} {}^\ast(X; \mathbb{Q})$.
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> 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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Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes
I have found a reference:



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> "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".
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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 $(X^+, \infty)$.


