GEOMETRICAL AND TOPOLOGICAL PROPERTIES OF SUBSPACES OF THE SPACE OF PROBABILITY MEASURES THAT ARE MANIFOLDS

Abstract

We study subspaces of the space  of probability measures  that are finite-dimensional and infinite-dimensional topological manifolds. Studying various properties of the subspaces of the space  of probability measures, the following are proved: for any closed subset  of the compactum  other than  itself, there exists a strong deformation retraction , for any infinite  and any of its closed subset  other than  itself, subspace  is barycentrically open, for any compact set  and for any  compact set  is  compact, for any infinite compact set  and any of its closed subset  other than , subspace  is homeomorphic to the Hilbert space , for any infinite compact set and any of its open subset  other than , the subspace  is homeomorphic to the Hilbert space , for any compact set and for any  the factor space  is  compact and the projection  is a homotopy equivalence, for any compact set  and for any  the following conditions are equivalent: a)  shape is dominated by some  compact;    b)  has a point shape; c)