On covering dimension and sections of vector bundles

An elementary result in point-set topology is used, with knowledge of the mod $2$ cohomology of real projective spaces, to establish classical results of Lebesgue and Knaster-Kuratowski-Mazurkiewicz, as well as the topological central point theorem of Karasev, which is applied to deduce results of Helly-Lovász, Bárány and Tverberg