Nearly Countable Dense Homogeneous Spaces

摘要

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n—1 such that S is invariant under all homeomorphisms of X and X S is countable dense homogeneous. We prove that every Borel space having fewer than c types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or c many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.