Continuous Cocycles Endowed with Point-Open Topology

Given a topological group $G$ and a Hausdorff topological group $A$ on which $G$ acts continuously and compatibly with the group operation of $A$, we study the set of continuous cocycles of $G$ with value in $A$. This set is a function space and can be endowed with several topologies. By imposing a suitable function space topology on the set of cocycles of $G$ with value in $A$, we propose a topological study of this set, and we prove, as our first main result, that if $A$ is a compact group having a presentation as an inverse limit of compact and Hausdorff topological groups $A_{r}$, for $r$ in a directed poset $R$, on which $G$ acts continuously and compatibly with the group operation of $A_{r}$ and equivariantly with respect to the transition maps, then one has a natural identification between the first nonabelian cohomology set of $G$ with coefficients in the inverse limit $A$ and the inverse limit of first nonabelian cohomology sets of $G$ with coefficients in $A_{r}$. Furthermore, we prove, as our second main result, that if $G$ is compact and Hausdorff, and $A$ is abelian - therefore one can define cohomology groups for all $n\geq1$ - then under a certain condition on $A_{r}$ one has a natural identification between cohomology group with coefficients in the inverse limit $A$ and the inverse limit of cohomology groups with coefficients in $A_{r}$, for all $n\geq1$.