A note on the Borsuk-Ulam theorem

Abstract

Let F \mathcal {F} denote the set of all maps from S n {S^n} to R n {{\mathbf {R}}^n} topologized by the usual metric, and B \mathcal {B} the set of all nonempty closed subsets of S n {S^n} invariant with respect to the antipodal map. Let β : F → B \beta :\mathcal {F} \to \mathcal {B} assign to each f ∈ F f \in \mathcal {F} the set of all x x for which f ( x ) = f ( − x ) f\left ( x \right ) = f\left ( { - x} \right ) . The largest topology on B \mathcal {B} for which β \beta is continuous is identified: it is the upper semifinite topology.