Continuity of Alexander-Spanier cohomology

Question

Suppose that a paracompact space $X$ is the inverse limit of paracompact spaces $X_{i}$ (that is $X=\varprojlim X_{i}$) and $H^{\ast }$ is Alexander-Spanier cohomology with closed supports. Then the equality $H^{\ast }\left( X,k\right) =\varinjlim H^{\ast }\left( X_{i},k\right) $ is true? (where $k$ is a field of characteristic zero.)

Answer 1

First of all, I am not sure if the below answers the question because of a terminological issue. I only know "closed support" in the context of Borel–Moore homology. For Alexander–Spanier cohomology, being defined in terms of functions where support is by definition closed, "closed support" seems to me to refer to the full complex where all functions are allowed – and this is what I am talking about in the below.

The Alexander–Spanier cohomology can be identified with Čech cohomology. On paracompact spaces, this is proved in Spanier's "Algebraic topology", Chapter 6, Section 8 "Fine presheaves". This actually holds for general spaces by a theorem of Dowker:

• C. Dowker. Homology groups of relations. Ann. Math. 56 (1952), 84–95.

So the continuity question for Alexander–Spanier cohomology is equivalent to the continuity question for Čech cohomology. However, in my comment I was a bit too hasty about the continuity for Čech cohomology. The classical references (e.g. Bredon's "Sheaf theory", Spanier's "Algebraic topology", or the original papers of Steenrod and Spanier) prove continuity for Čech cohomology on the category of compact Hausdorff spaces.

However, Čech cohomology satisfies a weak continuity property for paracompact Hausdorff spaces, by a result of Lee and Raymond:

• C.N. Lee and F. Raymond. Čech extensions of contravariant functors. Trans. Amer. Math. Soc. 133 (1968), 415–434. (paper available here)

Specifically, this is Theorem 5 in their paper. Note, however, that all the spaces in the inverse system are required to be embedded into one big space, with maps given by subspace inclusions. They define nested systems to additionally be those where the inverse limit is given by intersection and where every neighbourhood of the limit in the ambient space contains one of the spaces of the system. Weak continuity gives isomorphisms in cohomology only for inverse limits of nested systems. This is all discussed in Section 4 of the Lee–Raymond paper.

There's also a paper of Watanabe

• T. Watanabe. The continuity axiom and the Čech homology. Geometric topology and shape theory (Dubrovnik, 1986), 221–239, Lecture Notes in Math., 1283, Springer, 1987.

In the paper he discusses at length continuity properties of extensions of cohomology theories. If I'm reading his Corollary 22 right, it says that Čech cohomology is continuous on the category of pairs $(X,A)$ where X is a paracompact Hausdorff space with a closed subset $A$. Note, however, that there Čech cohomology is defined in terms of normal open coverings, i.e., those which have a partition of unity.