Images of the Cantor fan
type: Item
instance of: scholarly article
title: Images of the Cantor fan
zbMATH Open document ID: 0697.54019
DOI: 10.1016/S0166-8641(89)80005-7
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: A continuum \((=\) compact, connected, metric space) is hereditarily unicoherent if the intersection of any two subcontinua is connected. If it is also arcwise connected it is called a dendroid. If a dendroid has exactly one ramification point v, it is called a fan. The cone over the Cantor set is a fan which is called the Cantor fan \(F_ c\). The author is mainly concerned with characterizing the image of \(F_ c\) under confluent, open, monotone, retraction or light mappings. Five characterizations of the image of \(F_ c\) under a confluent mapping are given. For example, the image of \(F_ c\) is either an arc or a fan having the property of Kelley (a property similar to smoothness). Or, the image of \(F_ c\) is either an arc or a smooth fan whose set of end points union its ramification point v is closed. Similarly, six characterizations of the image of \(F_ c\) under an open mapping are given. For example, this image is either an arc or a smooth fan whose set of end points is closed. For monotone mappings there are two characterizations, one of which says: The nondegenerate continuum Y is the image of \(F_ c\) under a monotone mapping if and only if F is homomorphic either to \(F_ c\) or to \(F^ w_ c\). Finally, the author shows that the image of \(F_ c\) under a mapping (continuous, of course) is the image of \(F_ c\) under a light mapping.
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54C10
zbMATH DE Number: 4143014
zbMATH Keywords: dendroid
zbMATH Keywords: fan
zbMATH Keywords: cone over the Cantor set
zbMATH Keywords: Cantor fan
zbMATH Keywords: confluent mapping
zbMATH Keywords: property of Kelley
zbMATH Keywords: arc
zbMATH Keywords: smooth fan
zbMATH Keywords: ramification point
zbMATH Keywords: open mapping
zbMATH Keywords: monotone mappings
zbMATH Keywords: light mapping
author: Janusz J. Charatonik
author: Włodzimierz J. Charatonik
MaRDI profile type: Publication
cites work: Confluent mappings and unicoherence of continua
cites work: Q5548174
cites work: Fans with the property of Kelley
cites work: A note on smooth fans
cites work: Hyperspaces of a Continuum
cites work: Q4101154
cites work: Q5541833
cites work: Q3304890
cites work: On plane dendroids and their end points in the classical sense
cites work: Q5676808
cites work: Q4127048
cites work: Q5727237
full work available at URL: s0166-8641(89)80005-7
OpenAlex ID: W2109041610
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description: scientific article; zbMATH DE number 4143014