On n -thin dense sets in powers of topological spaces

Adam Bartoš

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 1, page 73-82
  • ISSN: 0010-2628

Abstract

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A subset of a product of topological spaces is called n -thin if every its two distinct points differ in at least n coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable T 3 space X without isolated points such that X n contains an n -thin dense subset, but X n + 1 does not contain any n -thin dense subset. We also observe that part of the construction can be carried out under MA.

How to cite

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Bartoš, Adam. "On $n$-thin dense sets in powers of topological spaces." Commentationes Mathematicae Universitatis Carolinae 57.1 (2016): 73-82. <http://eudml.org/doc/276795>.

@article{Bartoš2016,
abstract = {A subset of a product of topological spaces is called $n$-thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_3$ space $X$ without isolated points such that $X^n$ contains an $n$-thin dense subset, but $X^\{n + 1\}$ does not contain any $n$-thin dense subset. We also observe that part of the construction can be carried out under MA.},
author = {Bartoš, Adam},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {dense set; thin set; $\kappa $-thin set; independent family},
language = {eng},
number = {1},
pages = {73-82},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On $n$-thin dense sets in powers of topological spaces},
url = {http://eudml.org/doc/276795},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Bartoš, Adam
TI - On $n$-thin dense sets in powers of topological spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 1
SP - 73
EP - 82
AB - A subset of a product of topological spaces is called $n$-thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_3$ space $X$ without isolated points such that $X^n$ contains an $n$-thin dense subset, but $X^{n + 1}$ does not contain any $n$-thin dense subset. We also observe that part of the construction can be carried out under MA.
LA - eng
KW - dense set; thin set; $\kappa $-thin set; independent family
UR - http://eudml.org/doc/276795
ER -

References

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  1. Engelking R., General Topology, revised and completed edition, Sigma series in pure mathematics, 6, Heldermann, Berlin, 1989. Zbl0684.54001MR1039321
  2. Gruenhage G., Natkaniec T., Piotrowski Z., 10.1016/j.topol.2006.08.007, Topology Appl. 154 (2007), no. 4, 817–833. MR2294630DOI10.1016/j.topol.2006.08.007
  3. Hutchison J., Gruenhage G., 10.1016/j.topol.2011.07.005, Topology Appl. 158 (2011), no. 16, 2174–2183. MR2831904DOI10.1016/j.topol.2011.07.005
  4. Jech T., Set Theory, The Third Millennium Edition, revised and expanded, Springer, Berlin, 2002. Zbl1007.03002MR1940513
  5. Piotrowski Z., Dense subsets of product spaces, Questions Answers Gen. Topology 11 (1993), 313–320. MR1234206

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