Continuous cohomology group

Definition
Suppose $$G$$ and $$A$$ are topological groups (typically, taken to be T0 topological groups) such that $$A$$ is an abelian group and there is a topological group action of $$G$$ on $$A$$, i.e., a homomorphism $$\varphi:G \to \operatorname{Aut}(A)$$ such that the induced map $$G \times A \to A$$ is continuous. We can then define continuous cohomology groups $$H^n_\varphi(G;A)$$, each of which is an abelian group.

When $$\varphi$$ is understood from context, the subscript $${}_\varphi$$ may be omitted in the notation for the cohomology group, as well as the notation for the groups of $$n$$-cocycles and $$n$$-coboundaries.

Definition in terms of homological algebra
Note that these definitions are the same as those for the cohomology group in the discrete setting, with the following important difference: the $$\mathbb{Z}G$$-modules here are abelian topological groups with topological group actions of $$G$$ on them, and the morphisms between them must be continuous in addition to being equivariant under the $$G$$-action. Also, for the bar resolution, the cochains in the cochain complex comprise only the continuous functions.