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# On Countable Dense and $n$-homogeneity

Published:2012-03-05
Printed: Dec 2013
• Jan van Mill,
Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081a 1081HV, NETHERLANDS (NL)
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## 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 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
 MSC Classifications: 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]

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