Barriers in metric spaces

A. Dress, Vincent Moulton, Andreas Spillner, Taoyang Wu

Research output: Contribution to journalArticle

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Abstract

Defining a subset B of a connected topological space T to be a barrier (in T) if B is connected and its complement T-B is disconnected, we will investigate barriers B in the tight span

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of a metric D defined on a finite set X (endowed, as a subspace of RX, with the metric and the topology induced by the l8-norm) that are of the form
B=Be(f)?{g?T(D):?f-g?8=e}
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for some f?T(D) and some e=0. In particular, we will present some conditions on f and e which ensure that such a subset of T(D) is a barrier in T(D). More specifically, we will show that Be(f) is a barrier in T(D) if there exists a bipartition (or split) of the e-support View the MathML source of f into two non-empty sets A and B such that f(a)+f(b)=ab+e holds for all elements a?A and b?B while, conversely, whenever Be(f) is a barrier in T(D), there exists a bipartition of View the MathML source into two non-empty sets A and B such that, at least, f(a)+f(b)=ab+2e holds for all elements a?A and b?B.
Original languageEnglish
Pages (from-to)1150-1153
Number of pages4
JournalApplied Mathematics Letters
Volume22
Issue number8
DOIs
Publication statusPublished - Aug 2009

Keywords

  • Metric space
  • Tight span
  • Cutpoint

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