Are two different definitions for Čech cohomology equivalent?

Question

In Spanier's book Algebraic Topology (Chapter 6 section 7) he defines Čech cohomology in terms of the nerves of open coverings. I wish to know if this is equivalent, for a topological space A closed in X, to the direct limit of singular cohomologies of open sets in X containing A.

In particular, I am interested in the case where we have a constant coefficient group (rather than a presheaf) but for any topological space X (not necessarily Hausdorff or compact, etc.).

NOTE: I have reason to believe that they are not. In "Foundations of Algebra" (Eilenberg, Steenrod 1952), I believe that they define Čech cohomology in terms of nerves of open coverings, the same way that Spanier does. However, for this definition of Čech cohomology they show that the Eilenberg-Steenrod axioms hold, while the Eilenberg-Steenrod axioms don't necessarily hold for the alternative definition of Čech cohomology that I have described. Any clarifications or input welcome

EDIT: the second definition of Čech cohomology I've referenced is the one defined in Miller's "Lectures on Algebraic Topology" chapter 35 and is $\check{H}^n(A) := \varinjlim_{U \in \mathcal{U}_A} H^n(U)$ where $\mathcal{U}_A$ is the directed set whose elements are the open subsets of a topological space $X$ containing $A \subseteq X$ ordered such that $U \leq V$ when $V \subseteq U$

Answer 1

If $X$ is a closed subset of a manifold $M$, then the Cech cohomology of $X$ coincides with the direct limit of the singular cohomology of $U$ over open sets containing $X$. This becomes useful in discussing Poincare duality: one finds that (under compactness and orientation assumptions) $H_p(X,A)$ is isomorphic to the $(n-p)th$ Cech cohomology of $A$. But it is unfortunate to refer to this direct limit as a definition of Cech cohomology.

It seems that Miller is taking a short cut that has misled you; instead of giving the correct definition of Cech cohomology and stating that in this case it is isomorphic to that direct limit, he is simply referring to the direct limit as Cech cohomology.

(In a way, this is understandable. He has no need to say anything about the theory of Cech cohomology in order to prove theorems about duality in manifolds.)

Answer 2

The definition of Čech cohomology lends itself to a full sheaf cohomology theory over any fixed space or pair. When you have a sheaf cohomology you can test it against Grothendieck's axioms (as in his Tohoku paper from 1957). Singular cohomology leads to a sheaf cohomology generalization only for spaces and pairs that satisfy some additional condition, for example you may need local contractibility. It is for this restricted class of spaces that you can establish an isomorphism between the singular and Čech theories.

Answer 3

Let $\ X\ $ be a (Hilbert) cube of arbitrary dimension (finite or infinite -- of arbitrary cardinal number). Then the two cohomology theories of arbitrary closed subset $\ A\subseteq X\ $ (considered by OP) are canonically isomorphic.