The cantor set and inverse limits of upper semi-continuous functions

The cantor set and inverse limits of upper semi-continuous functions

In this paper, we study inverse limits with a single upper semi-continuous function F such that it is the union of mappings defined from a compact metric space X into itself. We prove that if Dom ( F ) is a totally disconnected space, then lim ⟵ ( X , F ) is homeomorphic to the Cantor set. This give...

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Bibliographic Details
Published in:Boletín de la Sociedad Matemática Mexicana Vol. 30; no. 3
Main Authors: Capulín, Félix, Ruiz del Portal, Francisco R., Sánchez-Garrido, Mónica
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01-11-2024
Springer Nature B.V
Subjects:
Continuity (mathematics)
Mathematics
Mathematics and Statistics
Metric space
Original Article
Cantor set
54B20
Upper semi-continuous functions
54B15
Inverse limits
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Description
Summary:In this paper, we study inverse limits with a single upper semi-continuous function F such that it is the union of mappings defined from a compact metric space X into itself. We prove that if Dom ( F ) is a totally disconnected space, then lim ⟵ ( X , F ) is homeomorphic to the Cantor set. This gives a partial answer to a problem posed by Ingram (Topol Appl 299:1–11, 2021), and it answers a question asked by Capulín et al. (Topol Proc 60:71–80, 2022).
ISSN:1405-213X
2296-4495
DOI:10.1007/s40590-024-00651-2