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Fixed Points can Characterise Curves

Published online by Cambridge University Press:  20 November 2018

Helga Schirmer
Helga Schirmer*
Affiliation:
Carleton University, Ottawa, CanadaK1S 5B6
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Abstract

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The concept of a firm fixed point of a selfmap of a metric space is introduced. Loosely speaking a fixed point is firm if it cannot be moved to a point nearby with the help of a map which is arbitrarily close to the given map. It is shown that a continuum always admits a selfmap with a firm fixed point if the continuum contains a triod and if the vertex of the triod has a neighbourhood which is a dendrite. This condition holds in particular for local dendrites. Hence a local dendrite is an arc or a simple closed curve if and only if it does not admit a selfmap which has a firm fixed point.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 2 , 01 June 1978 , pp. 207 - 211
Copyright
Copyright © Canadian Mathematical Society 1978

References

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