Nearly Countable Dense Homogeneous Spaces
Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 743-758
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We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n-1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\,\backslash \,S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.
Mots-clés :
54H05, 03E15, 54E50, countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture
Hrušák, Michael; Mill, Jan van. Nearly Countable Dense Homogeneous Spaces. Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 743-758. doi: 10.4153/CJM-2013-006-8
@article{10_4153_CJM_2013_006_8,
author = {Hru\v{s}\'ak, Michael and Mill, Jan van},
title = {Nearly {Countable} {Dense} {Homogeneous} {Spaces}},
journal = {Canadian journal of mathematics},
pages = {743--758},
year = {2014},
volume = {66},
number = {4},
doi = {10.4153/CJM-2013-006-8},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2013-006-8/}
}
TY - JOUR AU - Hrušák, Michael AU - Mill, Jan van TI - Nearly Countable Dense Homogeneous Spaces JO - Canadian journal of mathematics PY - 2014 SP - 743 EP - 758 VL - 66 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2013-006-8/ DO - 10.4153/CJM-2013-006-8 ID - 10_4153_CJM_2013_006_8 ER -
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