LET G, R AND A BE TOPOLOGICAL GROUPS AND (A,M) A PARTIALLY CROSSED TOPOLOGICAL G-R -BIMODULE. WE DEFINE THE SECOND NON-ABELIAN COHOMOLOGY, H2 (G,(A,M)), OF G WITH COEFFICIENTS IN (A,M). WE FIND A CRITERION UNDER WHICH H2 (G; (A, M)) ADMITS AN ABELIAN GROUP STRUCTURE. WE SHOW THAT THERE IS AN ACTION OF G ON H2 (G, (A; M)). MOREOVER, IF G IS A LOCALLY COMPACT HAUSDOR GROUP, THEN THE CENTER OF G ACTS TRIVIALLY ON H2 (G, (A, M)).