  In this paper, we introduce a symmetric continuous cohomology of topological
groups. This is obtained by topologizing a recent construction due to Staic (J.
Algebra 322 (2009), 1360-1378), where a symmetric cohomology of abstract groups
is constructed. We give a characterization of topological group extensions that
correspond to elements of the second symmetric continuous cohomology. We also
show that the symmetric continuous cohomology of a profinite group with
coefficients in a discrete module is equal to the direct limit of the symmetric
cohomology of finite groups. In the end, we also define symmetric smooth
cohomology of Lie groups and prove similar results.
