On confluently graph-like compacta
Volume 178 / 2003
Abstract
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.
