1. CMB 2012 (vol 56 pp. 860)
|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
2. CMB 2011 (vol 54 pp. 244)
|Homogeneous Suslinian Continua|
A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum $X$ has the property that the set of points at which $X$ is connected im kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.
Keywords:connected im kleinen, homogeneity, Suslinian, locally connected continuum
Categories:54F15, 54C05, 54F05, 54F50