Čech Cohomology with Coefficients in a Topological Abelian Group

Anordinary Čech cohomology (Formula Presented.) is defined for an arbitrary space X, and the group of coefficients G is assumed to be an Abelian group. On the category AC 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 (Formula Presented.) 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 (Formula Presented.) 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. © 2015, Springer Science+Business Media New York.