Let S and T denote complete
separable metric spaces. Let P(S) denote the collection of probability measures on S
and equip P(S) with the weak topology. If φ : S → T is continuous and onto, then φ
induces a weakly continuous mapping φ0 of P(S) onto P(T). We show that φ0 is
open in the weak topology if and only if φ is open. However, φ0 is always open in the
norm topology. Let K be a totally disconnected compact metric space and let SK
denote the set of continuous mappings of K into S. Then there exists a
natural mapping π of P(SK) into P(S)K. Blumenthal and Corson have
shown that π is onto. We establish that π is an open mapping in the weak
topology.
