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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A metric space with the Haver property whose square fails this property


Authors: Elzbieta Pol and Roman Pol
Journal: Proc. Amer. Math. Soc. 137 (2009), 745-750
MSC (2000): Primary 54D20, 54F45, 54E50
Published electronically: August 25, 2008
MathSciNet review: 2448597
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Abstract: Haver introduced the following property of metric spaces $ (X,d)$: for each sequence $ \epsilon _{1}, \epsilon _{2}, \ldots$ of positive numbers there exist collections $ \mathcal{V}_{1}, \mathcal{V}_{2}, \ldots$ of open subsets of $ X$, the union $ \bigcup _{i}\mathcal{V}_{i}$ of which covers $ X$, such that the members of $ \mathcal{V}_{i}$ are pairwise disjoint and every member of $ \mathcal{V}_{i}$ has diameter less than $ \epsilon _{i} $. We construct two separable complete metric spaces $ (X_{0},d_{0})$, $ (X_{1},d_{1})$ with the Haver property such that $ d_{0}$, $ d_{1}$ generate the same topology on $ X_{0}\cap X_{1}\neq \emptyset$, but $ (X_{0}\cap X_{1}, \max (d_{0},d_{1}))$ fails this property. In particular, the square of a separable complete metric space with the Haver property may fail this property. Our results answer some questions posed by Babinkostova in 2007.


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Additional Information

Elzbieta Pol
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Email: pol@mimuw.edu.pl

Roman Pol
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Email: pol@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09511-7
PII: S 0002-9939(08)09511-7
Keywords: Haver property, property $C$, product spaces.
Received by editor(s): September 24, 2007
Received by editor(s) in revised form: January 25, 2008
Published electronically: August 25, 2008
Additional Notes: The first author was partially supported by MNiSW Grant No. N201 034 31/2717
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society