  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-Lov\'asz, B\'ar\'any and Tverberg
