On confluently graph-like compacta
Fundamenta Mathematicae, Tome 178 (2003) no. 2, p. 109.

Voir la notice de l'article dans European Digital Mathematics Library

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.
Classification : 54F15, 54C10, 54C15, 54E45
Mots-clés : Confluent mapping, -mapping, confluently graph-like continua, confluently graph-representable continua 
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Lex G. Oversteegen; Janusz R. Prajs. On confluently graph-like compacta. Fundamenta Mathematicae, Tome 178 (2003) no. 2, p. 109. https://geodesic-test.mathdoc.fr/item/FUNDAM_2003__178_2_283219/