Nearly Countable Dense Homogeneous Spaces. (1st August 2014)

• Record Type: Journal Article
• Authors: Hrušák, Michael; van Mill, Jan
• Is Part Of: Canadian journal of mathematics. Volume 66:Number 4(2014)
• Page Start: 743
• Page End: 758
• Publication Date: 2014-08-01
• Subjects: 54H05 -- 03E15 -- 54E50; countable dense homogeneous -- nearly countable dense homogeneous -- Effros Theorem -- Vaught's conjecture; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2013-006-8
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: 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 $X\, \backslash \, S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.