A Bernstein parting of a space of measurable sets (Q808473)

Let Y be a separable metric space. A Bernstein parting of Y is a partition of Y into two connected dense subsets. The author proves several results about these partitions; for instance, let Y be a separable metric space such that for any a,b\(\in Y\), \(a\neq b\), there is a subspace P of Y (containing a and b) homeomorphic to the closed unit disc in \(R^ 2\), then Y has a Bernstein parting.