On confluently graph-like compacta

Lex G. Oversteegen ¹   ; Janusz R. Prajs ²

¹ Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294, U.S.A.
² Institute of Mathematics University of Opole Oleska 48 45-052 Opole, Poland and Department of Mathematics Idaho State University Pocatello, ID 83209, U.S.A.

Fundamenta Mathematicae, Tome 178 (2003) no. 2, pp. 109-127

Résumé

For any class ${\mathcal K}$ of compacta and any compactum $X$ we say that: (a) $X$ is confluently ${\mathcal K}$-representable if $X$ is homeomorphic to the inverse limit of an inverse sequence of members of ${\mathcal K}$ with confluent bonding mappings, and (b) $X$ is confluently ${\mathcal K}$-like provided that $X$ admits, for every $\varepsilon >0$, a confluent $\varepsilon $-mapping onto a member of ${\mathcal K}$. The symbol ${\mathbb L}{ \mathbb C}$ stands for the class of all locally connected compacta. It is proved in this paper that for each compactum $X$ and each family ${\mathcal K}$ of graphs, $X$ is confluently ${\mathcal K}$-representable if and only if $X$ is confluently ${\mathcal K}$-like. We also show that for any compactum the properties of: (1) being confluently graph-representable, and (2) being 1-dimensional and confluently ${ \mathbb L}{ \mathbb C}$-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.

Détail

DOI : 10.4064/fm178-2-2

Keywords: class mathcal compacta compactum say confluently mathcal representable homeomorphic inverse limit inverse sequence members mathcal confluent bonding mappings confluently mathcal provided admits every varepsilon confluent varepsilon mapping member mathcal symbol mathbb mathbb stands class locally connected compacta proved paper each compactum each family mathcal graphs confluently mathcal representable only confluently mathcal like compactum properties being confluently graph representable being dimensional confluently mathbb mathbb like equivalent consequently locally connected curves confluently graph representable conclude confluently arc like continua homeomorphic inverse limits arcs bonding mappings confluently tree like continua absolute retracts hereditarily unicoherent continua