Čech Cohomology with Coefficients in a Topological Abelian Group

Abstract

Anordinary Čech cohomology ${\overset{\smile }{H}}^{\ast}\left(X,G\right)$ is defined for an arbitrary space X, and the group of coefficients G is assumed to be an Abelian group. On the category A _C of compact pairs (X,A), an ordinary Čech cohomology satisfies the continuity axiom (see [1, Theorem 3.1.X]), i.e., we have the isomorphism

$$ {\overset{\smile }{H}}^{*}\left(X,A,G\right)\approx \underrightarrow{ \lim }{\overset{\smile }{H}}^{*}\left({X}_m,{A}_m,G\right), $$

where $\left(X,A\right)=\underleftarrow{ \lim}\left({X}_m,{A}_m\right),\left({X}_m,{A}_m\right)\in {A}_C$ Therefore, an ordinary Čech cohomology is called a continuous cohomology.

In the present paper, using a continuous singular cohomology (see [3]), we define a Čech cohomology ${\overset{\smile }{H}}^{\ast}\left(X,A,G\right)$ with coefficients in an arbitrary topological Abelian group G. We show that the defined cohomology satisfies the continuity axiom. This cohomology is investigated relative to a group of coefficients. In particular, given an inverse sequence of covering projections, a Čech cohomology with coefficients in the inverse limit of this sequence is isomorphic to the inverse limit of a sequence of Čech cohomologies in groups that are elements of the given sequence.