On Projectively Inductively Closed Subfunctors of the Functor $P$ of Probability Measures

In the paper, we examine topological and dimensional properties of metric, Tychonoff, compact $C$-spaces under the action of the covariant subfunctor $P_{f}$ of the functor $P$ of probability measures in the category of metric, compact, paracompact spaces and continuous mappings into itself. We consider geometric properties of spaces under the action of the subfunctor $P_{f}$ of the functor $P$ of probability measures and show that this functor $P_{f}$ is an open $\sigma$-p.i.c. functor that preserves soft mappings and various types of topological spaces.