Winding numbers are a basic notion in algebraic topology, and play an important role in connection with analytic functions of a complex variable. Intuitively, given a closed curve $t\mapsto S(t)$ in an oriented Euclidean plane (such as the complex plane $\mathbb{C}$ ), and a point $p$ not in the image of $S$ , the winding number (or index) of $S$ with respect to $p$ is the net number of times $S$ surrounds $p$ . It is not altogether easy to make this notion rigorous.

Let us take $\mathbb{C}$ for the plane. We have a continuous mapping $S:[a,b]\to \mathbb{C}$ where $a$ and $b$ are some reals with $a<b$ and $S(a)=S(b)$ . Denote by $\theta(t)$ the angle from the positive real axis to the ray from $z_0$ to $S(t)$ . As $t$ moves from $a$ to $b$ , we expect $\theta$ to increase or decrease by a multiple of $2\pi$ , namely $2\omega\pi$ where $\omega$ is the winding number. One therefore thinks of using integration. And indeed, in the theory of functions of a complex variable, it is proved that the value $$\frac{1}{2\pi i} \int_S \frac{dz}{z-z_0}$$ is an integer and has the expected properties of a winding number around $z_0$ . To define the winding number in this way, we need to assume that the closed path $S$ is rectifiable (so that the path integral is defined). An equivalent condition is that the real and imaginary parts of the function $S$ are of bounded variation.

But if $S$ is any continuous mapping $[a,b]\to \mathbb{C}$ having $S(a)=S(b)$ , the winding number is still definable, without any integration. We can break up the domain of $S$ into a finite number of intervals such that the image of $S$ , on any of those intervals, is contained in a disc which does not contain $z_0$ . Then $2\omega\pi$ emerges as a finite sum: the sum of the angles subtended at $z_0$ by the sides of a polygon.

Let $A$ , $B$ , and $C$ be any three distinct rays from $z_0$ . The three sets $$S^{-1}(A)\qquad S^{-1}(B)\qquad S^{-1}(C)$$ are closed in $[a,b]$ , and they determine the winding number of $S$ around $z_0$ . This result can provide an alternative definition of winding numbers in $\mathbb{C}$ , and a definition in some other spaces also, but the details are rather subtle.

For one more variation on the theme, let $S$ be any topological space homeomorphic to a circle, and let $f:S\to S$ be any continuous mapping. Intuitively we expect that if a point $x$ travels once around $S$ , the point $f(x)$ will travel around $S$ some integral number of times, say $n$ times. The notion can be made precise. Moreover, the number $n$ is determined by the three closed sets $$f^{-1}(a)\qquad f^{-1}(b)\qquad f^{-1}(c)$$ where $a$ , $b$ , and $c$ are any three distinct points in $S$ .