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      The groups \(S^3\) and \(SO(3)\) have no invariant binary \(k\)-network

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          Abstract

          A family \(\mathcal N\) of closed subsets of a topological space \(X\) is called a {\em closed \(k\)-network} if for each open set \(U\subset X\) and a compact subset \(K\subset U\) there is a finite subfamily \(\mathcal F\subset\mathcal N\) with \(K\subset\bigcup\F\subset \mathcal N\). A compact space \(X\) is called {\em supercompact} if it admits a closed \(k\)-network \(\mathcal N\) which is {\em binary} in the sense that each linked subfamily \(\mathcal L\subset\mathcal N\) is centered. A closed \(k\)-network \(\mathcal N\) in a topological group \(G\) is {\em invariant} if \(xAy\in\mathcal N\) for each \(A\in\mathcal N\) and \(x,y\in G\). According to a result of Kubi\'s and Turek, each compact (abelian) topological group admits an (invariant) binary closed \(k\)-network. In this paper we prove that the compact topological groups \(S^3\) and \(\SO(3)\) admit no invariant binary closed \(k\)-network.

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          Journal
          21 February 2011
          2013-11-04
          Article
          1102.4328
          b68aa97a-d756-438f-b149-c8e367303746

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          54D30, 22C05
          Topology Proc. 41 (2013) 261-266
          5 pages
          math.GN math.GR

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