Corollary 1

If T is a metrizable space of countable type, then $\mathscr{M}_+^b(T)$ is metrizable of countable type for the tight topology.

For, let $\hat{T}$ be the completion of T for a metric defining the topology of T; the space $\hat{T}$ is Polish, and $\mathscr{M}_+^b(T)$ is homeomorphic to the subspace of the Polish space $\mathscr{M}_+^b(\hat{T})$ consisting of the measures concentrated on T (No. 3, Prop. 8). But every subspace of a Polish space is metrizable of countable type (GT, IX, §2, No. 8).