Cech homology groups

Question

Let X and Y two compact Hausdorff spaces. If f is a homeomorphism of X into Y, then for each covering $\mathfrak{U}$ in $\Sigma(X)$, the collection $f(\mathfrak{U})$ of all images of elements of $\mathfrak{U}$ is an open covering of Y and conversely.

How can I prove that the Cech homology groups $H_{p}(X, G)$ and $H_{p}(Y, G)$ are isomorphic?

Thank you

Do not vandalize the question. – Pedro May 31 '13 at 18:08 — Pedro, May 31 '13 at 18:08
@PeterTamaroff read the edit history, what happened? – Olivier Bégassat May 31 '13 at 18:52 — Olivier Bégassat, May 31 '13 at 18:52

Norbert · Answer 1 · 2013-05-27T15:11:54.380

2

Note that $H_p(-,G):Top\to Ab$ is a functor.

Recall that homeomorphisms are isomorphisms in $Top$.

Finally prove (it is immediate) that any functor maps isomorphisms to isomorphisms.

edited May 27 '13 at 15:11

answered May 24 '13 at 07:52

Norbert

56,803

Is there another way to prove it without using homotopy? – May 24 '13 at 08:22
1

Where hadI used homotopy? – Norbert May 24 '13 at 08:37

Cech homology groups

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