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|On Countable Dense and $n$-homogeneity|
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 Problem 136 of Watson in the Open Problems in Topology Book in the negative.
Keywords:countable dense homogeneous, connected, $n$-homogeneous, strongly $n$-homogeneous, counterexample
Categories:54H15, 54C10, 54F05