The winding number of a closed curve around a point

TL;DR

This paper provides an elementary, rigorous definition of the winding number for closed curves around a point, simplifying its computation and illustrating its application to the Borsuk--Ulam theorem.

Contribution

It introduces a simpler, more direct definition of the winding number and demonstrates its use in elementary proofs of topological results.

Findings

Simplified method for defining the winding number

Easy computation techniques using additivity and intersection counting

Application to the Borsuk--Ulam theorem in low dimensions

Abstract

In this expository note we present an elementary direct rigorous definition and the simplest properties of the winding number. This definition is simpler than the one given in some textbooks. We show how to compute the winding number easily: using additivity or counting the (signed) intersection points. In the language of the winding number, we present an elementary formulation and proof of the low-dimensional case of the Borsuk--Ulam theorem. An English version is followed by a Russian version.