On confluently graph-like compacta

Lex G. Oversteegen, and Janusz R. Prajs. "On confluently graph-like compacta." Fundamenta Mathematicae 178.2 (2003): 109-127. <http://eudml.org/doc/283219>.

@article{LexG2003,
abstract = {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.},
author = {Lex G. Oversteegen, Janusz R. Prajs},
journal = {Fundamenta Mathematicae},
keywords = {Confluent mapping; -mapping; confluently graph-like continua; confluently graph-representable continua},
language = {eng},
number = {2},
pages = {109-127},
title = {On confluently graph-like compacta},
url = {http://eudml.org/doc/283219},
volume = {178},
year = {2003},
}

TY - JOUR
AU - Lex G. Oversteegen
AU - Janusz R. Prajs
TI - On confluently graph-like compacta
JO - Fundamenta Mathematicae
PY - 2003
VL - 178
IS - 2
SP - 109
EP - 127
AB - 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.
LA - eng
KW - Confluent mapping; -mapping; confluently graph-like continua; confluently graph-representable continua
UR - http://eudml.org/doc/283219
ER -