# On covering dimension and sections of vector bundles

## TL;DR

This paper uses elementary topology and cohomology to unify and extend classical theorems in topology and combinatorics, including Lebesgue, Knaster-Kuratowski-Mazurkiewicz, and Tverberg theorems.

## Contribution

It provides a new elementary approach to classical results in topology and combinatorics using mod 2 cohomology of real projective spaces.

## Key findings

- Unified proofs of classical theorems in topology and combinatorics
- Extension of the topological central point theorem
- New applications to Helly-Lovász, Bárány, and Tverberg results

## Abstract

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