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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on the Borsuk-Ulam theorem


Author: David Gauld
Journal: Proc. Amer. Math. Soc. 99 (1987), 571-572
MSC: Primary 54H25; Secondary 54C35, 54C60, 55M20
MathSciNet review: 875400
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Abstract: Let $ \mathcal{F}$ denote the set of all maps from $ {S^n}$ to $ {{\mathbf{R}}^n}$ topologized by the usual metric, and $ \mathcal{B}$ the set of all nonempty closed subsets of $ {S^n}$ invariant with respect to the antipodal map. Let $ \beta :\mathcal{F} \to \mathcal{B}$ assign to each $ f \in \mathcal{F}$ the set of all $ x$ for which $ f\left( x \right) = f\left( { - x} \right)$. The largest topology on $ \mathcal{B}$ for which $ \beta $ is continuous is identified: it is the upper semifinite topology.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0875400-X
Keywords: Borsuk-Ulam theorem, graph topology, upper semifinite topology
Article copyright: © Copyright 1987 American Mathematical Society