  Working over infinite dimensional separable Hilbert spaces, residual results
have been achieved for the space of contractive $C_{0}$-semigroups under the
topology of uniform weak operator convergence on compact subsets of
$\mathbb{R}_{+}$. Eisner and Ser\'eny raised in 2009 the open problem: Does
this space constitute a Baire space? Observing that the subspace of unitary
semigroups is completely metrisable and appealing to known density results, we
solve this problem positively by showing that certain topological properties
can in general be transferred from dense subspaces to larger spaces. The
transfer result in turn relies upon classification of topological properties
via infinite games. Our approach is sufficiently general and can be applied to
other contexts, e.g. the space of contractions under the pw-topology.
