The space of unitary $$C_{0}$$
C
0
-semigroups on a separable infinite-dimensional Hilbert space, when viewed under the topology of uniform weak operator convergence on compact subsets of $${\mathbb {R}}_{+}$$
R
+
, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [4]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive $$C_{0}$$
C
0
-semigroups constitute Polish spaces and thus positively addresses the open problem.
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