On Countable Dense and n-homogeneity

Abstract

Abstract.We prove that a connected, countable dense homogeneous space is n-homogeneous for every n, and strongly 2-homogeneous provided it is locally connected. We also present an example of a connected and countable dense homogeneous space which is not strongly 2-homogeneous. This answers in the negative Problem 136 ofWatson in the Open Problems in Topology Book.