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> If $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group.
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I hear that this is a classical result but I fail to see a clear proof in the literature. Dan Petersen says [here](https://mathoverflow.net/questions/254337/is-the-%C4%8Cech-cohomology-of-an-orbifold-isomorphic-to-its-singular-cohomology) that it is in Spanier's book. (I am assuming it is the Algebraic topology book by Spanier). I checked it twice roughly but somehow I could not find it. It would be a good idea to give some outline even if it is there.


Can someone give an outline of how this proof goes? There is some proof by using a **double cochain complex** [here](https://www.math.ru.nl/~mgroth/teaching/algtopII14/Section08.pdf). Is there a proof that does not use this double cochain complex set up? Please **do not** assume any familiarity with spectral sequences.



Outline of the proof that Cech cohomology and singular cohomology coincide on any locally contractible space
I think looking at Bredon's Sheaf theory book (page $179$ chapter $3$) would be helpful. 


It defines singular cohomology of a topological space $X$ with coefficients from **a sheaf of abelian groups** $\mathcal{A}$ by $H^k\_{Sing}(X,\mathcal{A})$ which boils down to (I did not check in detail but I am very much sure) usual definition of singular cohomology of a space $X$ with values in abelian group $A$ when $\mathcal{A}$ is just the sheaf associated to constant presheaf $U\mapsto A$ for each $U$ open in $X$ i.e.,
$$H^k\_{Sing}(X,\mathcal{A})\cong H^k\_{Sing}(X,A)$$
where right hand side is usual singular cohomology with coefficients from $A$ and $\mathcal{A}$ is the sheaf associated to constant presheaf defined as $U\mapsto A$ for each $U$ open in $X$.


This way of seeing gives **less surprise** and **more clarity** for me. 


It might be **surprising** to see that 



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> On a paracompact space $X$, there is an isomorphism $H^k\_{Sing}(X,A)\cong H^k\_{Sheaf}(X,\underline{A}\_X)$ where $\underline{A}\_X$ is the sheaf associated to constant presheaf defined as $U\mapsto A $ for all $U$ open in $X$.
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It is definitely little **less surprising** and **more natural** to see that 



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> On a paracompact space $X$, there is an isomorphism $H^k\_{Sing}(X,\mathcal{A})\cong H^k\_{Sheaf}(X, \mathcal{A})$ for any sheaf of abelian groups $\mathcal{A}$ on $X$.
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This proof is in Bredon's book. I do not prove that result but I mentioned just to give some motivation.


To define sheaf cohomology $H^k\_{Sing}(X,\underline{A}\_X)$ we need to look for a injective resolution of $\underline{A}\_X$ and apply Global section functor to get Sheaf cohomology $H^k\_{Sing}(X,\underline{A}\_X)$.


Ramanan's Global calculus book (page $105$ lemma $3.1,3.2$) says 



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> Let $0\rightarrow \mathcal{F}\rightarrow \mathcal{G}^\bullet$ be **an arbitrary** resolution of the $\mathcal{F}$. Suppose $H^i(X,\mathcal{G}^j)=0$ for each $i>0$ and $j\geq 0$. Then the complex $\mathcal{G}^\bullet$ has cohomologies canonically isomorphic to those of $\mathcal{F}$ i.e.,$H^k(\mathcal{G}^\bullet(X))$ is isomorphic to $H^k(X,\mathcal{F})$ for each $k>0$.
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This result is relavent here because the idea is to produce a resolution of $\underline{A}\_X$ that has the above property and we use that resolution to produce sheaf cohomology groups $H^k(X,\underline{A}\_X)$.


Given an open set $U$ of $X$, consider singular $n$-cochains 
inside $U$ i.e., $\alpha:\{\Delta\_n\rightarrow U\}\rightarrow A$. Here, $\Delta\_n$ is the standard$\alpha:\{\Delta\_n\rightarrow U\}\rightarrow A$ $n$-simplex and $\Delta\_n\rightarrow U$ is $n$-singular simplex in $U$ and $\alpha:\{\Delta\_n\rightarrow U\}\rightarrow A$ is $n$-singular cochain in $U$ with values in $A$. The collection of all such $\alpha$ is denoted by $S^k(U)$. This is an abelian group. $\alpha:\{\Delta\_n\rightarrow U\}\rightarrow A$


The assignment $U\rightarrow S^k(U)$ gives a presheaf $S^k$ of abelian groups on $X$. The sheaf associated to this presheaf is denoted by $\widetilde{S^k}$ on $X$ for each $X$. We get a resolution
$$0\rightarrow \mathcal{F}\rightarrow \widetilde{S^0}\rightarrow \widetilde{S^1}\rightarrow\cdots$$
In case when $X$ is **paracompact** the sheaf $\widetilde{S^k}$ is flabby sheaf (also called as flasque sheaf) for each $k$. For every flabby sheaf $\mathcal{G}$ we have $H^i(X,\mathcal{G})=0$ for each $i>0$. Thus, we can use this resolution $$0\rightarrow \widetilde{S^0}
\rightarrow \widetilde{S^1}\rightarrow \cdots$$
Taking global section functor we have 
$$0\rightarrow \widetilde{S^0}(X)
\rightarrow \widetilde{S^1}(X)\rightarrow \cdots$$
As $\widetilde{S^k}(X)$ is related to presheaf of singular $n$-cochains on $X$, it is only expected that this cochain complex has same cohomology groups as that of 
$$0\rightarrow \mathcal{C}^0(X)\rightarrow \mathcal{C}^1(X)\rightarrow $$
where $\mathcal{C}^i(X)$ is just the singular $n$-cochains on $X$ with values in $A$. Cohomology groups of this cochain complex is precisely the singular cohomology groups $H^k(X,A)$. Thus, sheaf cohomology groups $H^k(X,\underline{A}\_X)$ are isomorphic to singular cohomology groups $H^k(X,A)$.


As the sheaf cohomology $H^k(X,\underline{A}\_X)$ agrees with Cech cohomology $H^k(\mathcal{U},\underline{A}\_X)$ when we have good cover $\mathcal{U}$. As $X$ is **locally contractible** that cover that we choose from a line bundle can be choosen to be good cover. In that case, we have $H^k (\mathcal{U},\underline{A}\_X)$ is canonically isomorphic to $H^k (\mathcal{U},\underline{A}\_X)$.


Thus, comparing above two results, we see that 
$$H^k\_{\text{Cech}}(\mathcal{U},\underline{A}\_X)\cong H^k\_{\text{Sheaf}}(X,\underline{A}\_X)\cong H^k\_{\text{Sing}}(X,A)$$


