Star-shaped plane domains are homologically simply connected

Given: A nonempty open $U\subseteq \mathbb{C}$ star-shaped with respect to $a\in U$; the plane identification and its segments are those of Complex star-shaped and convex domains are the published Euclidean notions under the identification $\mathbb{C}={\mathbb{R}}^2$.

A complex domain is homologically simply connected when every cycle with trace in it is null-homologous in it (Homologically simply connected complex domains), and a cycle $\mathrm{\Gamma }$ with trace in $\mathrm{\Omega }$ is null-homologous in $\mathrm{\Omega }$ when $n(\mathrm{\Gamma },p)=0$ for every $p\in \mathbb{C}\setminus \mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

If $\mathrm{\Gamma }$ is a cycle whose trace lies in an open $V$ and $F$ is a primitive on $V$ of a continuous $f$ with $F^{\mathrm{\prime }}=f$ continuous, then ${\int }_{\mathrm{\Gamma }}fdz=0$ (The integral of a continuous derivative over a cycle is zero).

If $U\subseteq \mathbb{C}$ is open and star-shaped with respect to $a\in U$, every holomorphic $f:U\to \mathbb{C}$ has the primitive $F(z)={\int }_{{\mathrm{\ell }}_{az}}f(\zeta )d\zeta$ (Every holomorphic function on a star-shaped domain has a primitive).

A nonempty open $U\subseteq {\mathbb{R}}^n$ is star-shaped with respect to $a\in U$ when $a+t(x-a)\in U$ for every $x\in U$ and $0\le t\le 1$; every convex open set is star-shaped with respect to each of its points (Star-shaped open subsets of Euclidean space, A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points).

Constants and the identity are complex differentiable, and linear combinations, products and nonvanishing quotients of functions complex differentiable at a point are complex differentiable there (Linearity, product, reciprocal, and quotient rules for complex derivatives); a complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

$n(\mathrm{\Gamma },p)=(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}dz\mathrm{/}(z-p)$ for a chain $\mathrm{\Gamma }$ and $p\mathrm{\notin }{\mathrm{\Gamma }}^{\ast }$ (Integration over a complex chain and the index of a chain), a chain being a finite list of integer-weighted contours (Complex chains, their traces, and cycles).

A complex domain is a nonempty, connected, open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$).

A subset is path-connected when any two of its points are joined by a continuous map from $[0,1]$ with image inside it (Paths, path-connected spaces and path components), and a path-connected subset is connected (Every path-connected space is connected, and every path component lies inside a component); a composite of continuous maps is continuous and a function continuous on each member of a finite closed cover is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

A primitive of $f$ on $V$ is a holomorphic $F$ with $F^{\mathrm{\prime }}=f$ on $V$ (A primitive of a complex function on an open set).