Mappings and Prohorov spaces
by
Mitrofan Cioban
Academy of Sciences of Moldova

On the category of Tichonoff spaces consider the covariant functor P, where P(X) is the space of all Radon probability measures on X with the topology of weak convergence.

Let A and B be classes of topological spaces and C be a class of continuous mappings, f:X --> Y be a surgection, X in A, and Y in B and f in C. We consider the following general questions:

P1. Let X be a Prohorov space. Is it true that P(X) in A? Is it true that P(X) is a Prohorov space? Is it true that X or P(X) is a Baire space?

P2. Let X be a Prohorov space. Is it true that Y is a Prohorov space?

P3. Let Y be a Prohorov space. Is it true that X is a Prohorov space?

P4. Is it true that P(f) in C?

A space is sieve-complete if it is a continuous image of some Cech complete space. A mapping f:X --> Y is uniformly complete if there exists a sieve-complete extension Z of X such that the fibers f^-1(y) are closed in Z. A space Y is an A(k)-space iff there exist a paracompact p-space X and an open continuous uniformly complete surjection f:X --> Y.

In particular, the following assertions are proved:

A1. A space X is sieve-complete iff P(X) is sieve-complete.

A2. A space X is an A(k)-space iff P(X) is an A(k)-space.

A3. Let X be a Prohorov space of pointwise countable type. Then X is a Baire space.

A4. Let f:X --> Y be an open continuous uniformly complete mapping of an A(k)-space X onto Y. Then X is a Prohorov space iff Y is Prohorov. Moreover, the mapping P(f) is open and uniformly complete.

A5. There exist three regular countable spaces X, Y, Z and two continuous open finite-to-one surjections f:X --> Y, g:Y --> Z such that X is Prohorov, Z is compact and Y is not a Prohorov space.

A6. The two-arrow compact space of Alexandroff-Urysohn is a first countable Prohorov space which is not an open continuous image of some metrizable Prohorov space.

Some open questions will be formulated.

Date received: May 31, 2003