A note on the Borsuk-Ulam theorem

Gauld, David

doi:10.1090/S0002-9939-1987-0875400-X

A note on the Borsuk-Ulam theorem
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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.

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