caa a22 4500 60552307X CHVBK 20210128100748.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s10958-015-2601-4 doi (NATIONALLICENCE)springer-10.1007/s10958-015-2601-4 Čech Cohomology with Coefficients in a Topological Abelian Group [Elektronische Daten] [L. Mdzinarishvili, L. Chechelashvili] Anordinary Čech cohomology H ⌣ ∗ X G $$ {\overset{\smile }{H}}^{\ast}\left(X,G\right) $$ 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 H ⌣ * X A G ≈ lim → H ⌣ * X m A m G , $$ {\overset{\smile }{H}}^{*}\left(X,A,G\right)\approx \underrightarrow{ \lim }{\overset{\smile }{H}}^{*}\left({X}_m,{A}_m,G\right), $$ where X A = lim ← X m A m , X m A m ∈ A C $$ \left(X,A\right)=\underleftarrow{ \lim}\left({X}_m,{A}_m\right),\left({X}_m,{A}_m\right)\in {A}_C $$ 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 H ⌣ ∗ X A G $$ {\overset{\smile }{H}}^{\ast}\left(X,A,G\right) $$ 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. Springer Science+Business Media New York, 2015 Mdzinarishvili L. Georgian Technical University, Tbilisi, Georgia aut Chechelashvili L. Georgian Technical University, Tbilisi, Georgia aut Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 211/1(2015-11-01), 40-57 1072-3374 211:1<40 2015 211 10958 https://doi.org/10.1007/s10958-015-2601-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-015-2601-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Mdzinarishvili L. Georgian Technical University, Tbilisi, Georgia aut NATIONALLICENCE 700 1- Chechelashvili L. Georgian Technical University, Tbilisi, Georgia aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 211/1(2015-11-01), 40-57 1072-3374 211:1<40 2015 211 10958