# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A metric space with the Haver property whose square fails this propertyHTML articles powered by AMS MathViewer

by Elżbieta Pol and Roman Pol
Proc. Amer. Math. Soc. 137 (2009), 745-750 Request permission

## 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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• Elżbieta 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
• 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