A note on the Borsuk-Ulam theorem
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- by David Gauld PDF
- Proc. Amer. Math. Soc. 99 (1987), 571-572 Request permission
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.References
- Jan Jaworowski, A continuous version of the Borsuk-Ulam theorem, Proc. Amer. Math. Soc. 82 (1981), no. 1, 112–114. MR 603612, DOI 10.1090/S0002-9939-1981-0603612-3
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- Herbert Robbins, Some complements to Brouwer’s fixed point theorem, Israel J. Math. 5 (1967), 225–226. MR 221494, DOI 10.1007/BF02771610
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 99 (1987), 571-572
- MSC: Primary 54H25; Secondary 54C35, 54C60, 55M20
- DOI: https://doi.org/10.1090/S0002-9939-1987-0875400-X
- MathSciNet review: 875400