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 countable dense homogeneous, connected, $n$-homogeneous, strongly $n$-homogeneous, counterexample

MSC Classifications:

54H15, 54C10, 54F05 show english descriptions Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]
Special maps on topological spaces (open, closed, perfect, etc.)
Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces [See also 06B30, 06F30] 54H15 - Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]
54C10 - Special maps on topological spaces (open, closed, perfect, etc.)
54F05 - Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces [See also 06B30, 06F30]

top of page | contact us | social media | privacy | site map