On Topological Structure of the First Non-abelian Cohomology of Topological Groups

Let G, R, and A be topological groups. Suppose that G and R act continuously on A, and G acts continuously on R. In this paper, we dene a partially crossed topological G - R-bimodule (A, μ), where μ: A→ R is a continuous homomorphism. Let Derc(G, (A, μ)) be the set of all (α, r) such that α : G → A is a continuous crossed homomorphism and μα(g) = rgr-1. We introduce a topology on Derc(G, (A, μ)). We show that Derc(G, (A, μ)) is a topological group, wherever G and R are locally compact. We define the first cohomology, H1(G, (A, μ)), of G with coecients in (A, μ)) as a quotient space of Derc(G, (A, μ)). Also, we state conditions under which H1(G, (A, μ)) is a topological group. Finally, we show that under what conditions H1(G, (A, μ)) is one of the following: k-space, discrete, locally compact and compact.