Winding number
Formal definition

Let γ: → C ∖ { a } {\displaystyle \gamma :\to \mathbb {C} \setminus \{a\}} be a continuous closed path on the plane minus one point. The winding number of γ {\displaystyle \gamma } around a {\displaystyle a} is the integer

where (ρ, s) {\displaystyle (\rho,s)} is the path written in polar coordinates, i.e. the lifted path through the covering map

The winding number is well defined because of the existence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of p {\displaystyle p} are of the form ρ 0 × (s 0 + Z) {\displaystyle \rho _{0}\times (s_{0}+\mathbb {Z})} (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed.
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