Subspaces dimensional properties that are boundary sets of the probability measures space, defined in an infinite compactum X

In this note, we consider dimensional properties of the subspace of probability measure spaces P(X) for which transfinite dimensional functions ind, Ind и dim are defined. It is shown that countability of a compact set X is equivalent to the existence of dimensions indPω(X) , IndPω(X), dimPω(X), indPf(X), IndPf(X) и dimPf(X) for the subspaces Pω(X), Pf(X), Pn(X) respectively. It is also noted that for any compact C-space of the subspaces Pn(X), Pω(X), Pf(X) the space P (x) are compact C-spaces. If for an infinite compact set X the subspace Pω(X) contains the Hilbert cube Q, then there exists a number n∈N, n>1, such that Xn x σn-1 contains the Hilbert cube Q. Further, for an infinite compact set X, a number of subspaces Y of the compact set P (x) which are Q-, fj-, k{- and E-manifolds are identified. In particular, for a proper closed subset AcX, the subspaces Sp(A) есть l2 for any n ∈ N (n>1), P(X) \ Pn(X) are Q-manifolds; for any proper everywhere dense countable subspace A⊂X, the subspace Pω(A) is the boundary set of the compact set P (x). If Pω(X) contains the Hilbert cube Q, then the subspace Pω(X) is homeomorphic to the space ∑. It is considered in which cases everywhere dense subsets A of the spaces P(X) defined in an infinite compactum X are its boundary set. It is also shown which everywhere dense subsets A⊂P(X) and B⊂P(Y) for infinite compact sets X and Y of the spaces P(X) and P(Y), respectively, are at the same time mutually homeomorphic.

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