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Smallest weakly contractible non-contractible topological spaces

Published online by Cambridge University Press:  11 December 2019

Nicolás Cianci
Affiliation:
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, Mendoza, Argentina (mottina@fcen.uncu.edu.ar)
Miguel Ottina
Affiliation:
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, Mendoza, Argentina (mottina@fcen.uncu.edu.ar)
Corresponding
E-mail address:

Abstract

We characterize the topological spaces of minimum cardinality which are weakly contractible but not contractible. This is equivalent to finding the non-dismantlable posets of minimum cardinality such that the geometric realization of their order complexes are contractible. Specifically, we prove that all weakly contractible topological spaces with fewer than nine points are contractible. We also prove that there exist (up to homeomorphism) exactly two topological spaces of nine points which are weakly contractible but not contractible.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2019

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