On the equivalence of Singular, Alexander-Spanier and Cech cohomology (with compact support) It is known that Singular cohomology $H^*$, Alexander-Spanier cohomology $\bar H^*$ and Cech cohomology $\check H^*$ all agree over compact simplicial complexes and therefore on compact (smooth) manifolds.
Moreover $\bar H^*$ and $\check H^*$ coincide  over compact and Hausdorff spaces.
My question regards a generalisations of this.


*

*X is a non-compact simplicial complex do we still have  that $H^*(X)$, $\bar H^*(X)$ and $\check H^*(X)$ all agreee?

*Does the same apply to $H_c^*(X)$, $\bar H_c^*(X)$ and $\check H_c^*(X)$ cohomology with compact support?  Is there something that plays the role of Eilenberg-Steenrod axioms in this case?


 A: This is only a partial answer.
In

Dowker, Clifford H. "Homology groups of relations." Annals of mathematics (1952): 84-95.

one can find a proof that Alexander-Spanier cohomology and Cech cohomology agree for all pairs (see Theorem 2).
Wikipedia says

The Alexander–Spanier cohomology groups coincide with Cech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.

But in fact Cech cohomology groups agree with singular cohomology groups  for all CW-complexes. Here are some references.

*

*In
Morita, Kiiti. "Čech cohomology and covering dimension for topological spaces." Fundamenta Mathematicae 1.87 (1975): 31-52.
one finds a variant of Cech cohomology based on normal open covers instead of arbitrary open covers. In paracompact spaces all open covers are normal, thus the "usual" Cech cohomology groups agree with the "Morita-Cech" cohomology groups for paracompact spaces.


*In
Mardešic, Sibe, and Jack Segal. Shape theory: the inverse system approach. Elsevier, 1982.
Morita's approach is taken up in modified form (Chapter II §3.2). The Morita-Cech cohomology groups of a space $X$ are defined as $\check H^n(X) = \varinjlim H^n(\mathbf X)$, where the inverse system $\mathbf X$ occurs in a so-called $HPOl$-expansion $\mathbf p : X \to \mathbf X$. One can take for example the Cech-expansion (based on normal coverings). If $X$ has the homotopy type of a polyhedron (wich is true for CW complexes and ANRs), then one can take the trivial $HPOl$-expansion consisting of the trivial inverse system $\mathbf X= (X)$ and $\mathbf p = id$. We see that $\check H^n(X)  = H^n(X)$. Now observe that CW complexes are paracompact.
