The space of contractive C0-semigroups is a Baire space

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C0-semigroups under topology uniform weak operator convergence on compact subsets R+. Eisner and Serény raised in [6] [3] open problem: Does this constitute a Baire space? Observing that subspace unitary semigroups is completely metrisable appealing to known density results, we solve problem positively by showing certain topological properties can general be transferred from dense subspaces larger spaces. The transfer result turn relies upon classification via games. Our approach sufficiently applied other contexts, e.g. contractions -topology.