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Monotone Classes of Dendrites

Published:2015-06-03
Printed: Jun 2016
• Veronica Martínez-de-la-Vega,
Instituto de Matemáticas, Universidad Nacional Autónoma de México Circuito exterior, Cd. Universitaria, México D.F., 04510 México
• Christopher Mouron,
Department of Mathematics and Computer Science, Rhodes College, Memphis, TN 38112, USA
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Abstract

Continua $X$ and $Y$ are monotone equivalent if there exist monotone onto maps $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$. A continuum $X$ is isolated with respect to monotone maps if every continuum that is monotone equivalent to $X$ must also be homeomorphic to $X$. In this paper we show that a dendrite $X$ is isolated with respect to monotone maps if and only if the set of ramification points of $X$ is finite. In this way we fully characterize the classes of dendrites that are monotone isolated.
 Keywords: dendrite, monotone, bqo, antichain
 MSC Classifications: 54F50 - Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03] 54C10 - Special maps on topological spaces (open, closed, perfect, etc.) 06A07 - Combinatorics of partially ordered sets 54F15 - Continua and generalizations 54F65 - Topological characterizations of particular spaces 03E15 - Descriptive set theory [See also 28A05, 54H05]

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