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2003 | 178 | 2 | 109-127
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On confluently graph-like compacta

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For any class 𝒦 of compacta and any compactum X we say that: (a) X is confluently 𝒦-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of 𝒦 with confluent bonding mappings, and (b) X is confluently 𝒦-like provided that X admits, for every ε >0, a confluent ε-mapping onto a member of 𝒦. The symbol 𝕃ℂ stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family 𝒦 of graphs, X is confluently 𝒦-representable if and only if X is confluently 𝒦-like. We also show that for any compactum the properties of: (1) being confluently graph-representable, and (2) being 1-dimensional and confluently 𝕃ℂ-like, are equivalent. Consequently, all locally connected curves are confluently graph-representable. We also conclude that all confluently arc-like continua are homeomorphic to inverse limits of arcs with open bonding mappings, and all confluently tree-like continua are absolute retracts for hereditarily unicoherent continua.
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  • Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, U.S.A.
  • Institute of Mathematics, University of Opole, Oleska 48, 45-052 Opole, Poland
  • Department of Mathematics, Idaho State University, Pocatello, ID 83209, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-2-2
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