On euclidean manifolds being a subspace of the space of probability measures with finite supports to a certain infinite compact set of dimension zero

In this short communication we prove that the subspace $P_{n,n-1} (X)$ of all probability measures $P(X)$, whose supports consist of exactly $n$ points is an $(n-1)$-dimensional topological manifold. A number of subspaces of the space of all probability measures having infinite dimension in the sense of dim, which are manifolds, are identified. We also consider individual subsets of the infinite compact set $\mathrm{X}$, on which the space of probability measures is homotopy dense in the entire space. Three theorems on the topological properties of manifolds—subspaces of homotopy dense probability measures in the space of probability measures with finite supports on a compactum—are formulated and proven, and special cases of finite and infinite compactums are considered.