The topological support of the z-measures on the Thoma simplex

The Thoma simplex $\Omega$ is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are a family of probability measures on $\Omega$ depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit when it goes to $0$, the z-measures turn into the Poisson-Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space $\Omega$. The proof is based on results of arXiv:0902.3395 and arXiv:1806.07454.