The winding number is constant on each connected component of the complement of the trace

Given: A closed complex contour $\gamma :[a,b]\to \mathbb{C}$; the plane carries the Euclidean metric of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves.

For a closed complex contour $\gamma$ and $p\mathrm{\notin }{\gamma }^{\ast }$, $n(\gamma ,p)=(2\pi i{)}^{-1}{\int }_{\gamma }dz\mathrm{/}(z-p)$ (The winding number of a closed contour about a point off its trace).

The winding number of a closed complex contour about a point off its trace is an integer (The winding number of a closed contour is an integer).

For a complex contour $\gamma$ and $p_0\mathrm{\notin }{\gamma }^{\ast }$, the distance $d=\inf \{\mathrm{\mid }w-p_0\mathrm{\mid }:w\in {\gamma }^{\ast }\}$ exists and is positive (A contour missing a point subdivides into arcs lying in discs that miss it).

If $\mathrm{\mid }f(z)\mathrm{\mid }\le M$ on the trace of a rectifiable contour $\gamma$, with $M\ge 0$, then $\mathrm{\mid }{\int }_{\gamma }f(z)dz\mathrm{\mid }\le ML(\gamma )$ (ML estimate: a contour integral is bounded by a supremum bound times path length).

For continuous $f,g$ on the trace of a rectifiable contour $\gamma$ and $\alpha ,\beta \in \mathbb{C}$, ${\int }_{\gamma }(\alpha f+\beta g)dz=\alpha {\int }_{\gamma }fdz+\beta {\int }_{\gamma }gdz$ (Complex line integrals are linear in the integrand).

The connected component $C(x)$ is the union of all connected subsets containing $x$, hence the largest connected subset containing $x$ (Connected components, quasicomponents, and totally disconnected spaces).

Every connected component of an open subset $U\subseteq {\mathbb{R}}^n$ is open in ${\mathbb{R}}^n$ and polygonally connected (Every connected component of an open subset of ${\mathbb{R}}^n$ is open and polygonally connected).

The continuous image of a connected subset is connected (A continuous image of a connected space is connected, and connectedness is a topological property), and a connected subset of $\mathbb{R}$ is order-convex (The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$").

A compact subset of a metric space is closed and bounded (A compact subset of a metric space is closed and bounded); the continuous image of a compact subset is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); and a closed bounded interval is compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

A set is closed exactly when its complement is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), and $B(x,r)=\{y:d(x,y)<r\}$ (Open ball, closed ball and sphere in a metric space).

$\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$ and $\mathrm{\mid }z+w\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }$ for complex $z,w$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

The integers form an ordered commutative ring, and their canonical image in $\mathbb{R}$ is discrete; hence if $m<n$ then $m+\frac{1}{2}$ lies strictly between them and is not an integer (The integers form a commutative ring, The integers form a totally ordered ring, Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).