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Strong colorings yield $ \kappa$-bounded spaces with discretely untouchable points


Authors: István Juhász and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 143 (2015), 2241-2247
MSC (2010): Primary 54A25, 03E05, 54D30
DOI: https://doi.org/10.1090/S0002-9939-2014-12394-X
Published electronically: December 22, 2014
MathSciNet review: 3314130
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Abstract: It is well known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklóssy and the first author, we show that this statement fails for countably compact regular spaces, and even for $ \omega $-bounded regular spaces. In fact, there are $ \kappa $-bounded counterexamples for every infinite cardinal $ \kappa $. The proof makes essential use of the so-called strong colorings that were invented by the second author.


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Additional Information

István Juhász
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 13-15 Reáltanoda u. 1053 Budapest, Hungary
Email: juhasz@renyi.hu

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Har ha-Tsofim, Jerusalem, Israel
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9939-2014-12394-X
Keywords: Strong colorings, discretely untouchable points, $\kappa$-bounded spaces
Received by editor(s): June 25, 2013
Received by editor(s) in revised form: October 14, 2013
Published electronically: December 22, 2014
Additional Notes: The first author was partially supported by OTKA grant no. K 83726
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2014 American Mathematical Society

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