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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-2-2,
author = {Lex G. Oversteegen and Janusz R. Prajs},
title = {On confluently graph-like compacta},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {109-127},
zbl = {1054.54010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-2-2}
}
Lex G. Oversteegen; Janusz R. Prajs. On confluently graph-like compacta. Fundamenta Mathematicae, Tome 177 (2003) pp. 109-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-2-2/