Uma nota sobre o teorema de Borsuk-Ulam

Abstract

The main objective of this work is to prove that the map B defined on F and taking values in B, where F is the set of all continuous functions from Sn to Rn and B is the set of all nonempty closed subsets of Sn, invariant under the antipodal map, which assign to each f 2 F the set fx 2 Sn; f(x) = f(��x)g, is continuous when the topology of F is the topology induced by the usual metric, and the topology of B is the upper semi-finite topology. Considering in F the topology induced by the usual metric, we will have that the finest topology in B such that the map B is continuous is the upper semi-finite topology.