3.4 Connectedness
Definition 3.30. A set \(S\subseteq \mathbb {C}\) is said to be connected if its not contained in the union of two disjoint
nonempty open subsets of \(\mathbb {C}\) which have a non trivial intersection with \(S\)
\(\big [ G_1 \cup G_2,\quad G_1\neq \emptyset , \quad G_1\cap G_2 = \emptyset \quad G_1\) and \(G_2\) are open. \(\, G_1\cap S \neq \emptyset ,\quad G_2\cap S \neq \emptyset \big ]\)
An open connected nonempty subset of \(\mathbb {C}\) is called a region or a domain.
Proposition 3.31. Let \(G\) be a nonempty open subset of \(\mathbb {C}\). Then \(G\) is a region if and only i any two points of \(G\) can be connected by a polygonal path.
Proof.
Suppose that \(G\) is a region. Fix \(a\in G\) and let \(G_1 = \big \{z\in G:\,\) there is a polygonal path from \(a\) to \(z\) in \(G\big \}\).
Let \(G_2 = G/G_1\, (G_1\cup G_2 = G \implies \) either \(G_1 = \emptyset \) or \(G_2 = \emptyset )\).
Suppose \(z\in G\), then since \(G_1\subset G\) and \(G\) is open then there is an \(\, r> 0 \ni B(z;r)\subset G_1\). Since \(a\) and \(z\) are connected by a polygonal
path, we can extended this path to any point \(\, w \in B(z; r)\,\) by joining an additional segment joining \(z\) and \(w\).
So \(\,w \in B(z; r)\,\) is such that \(\, w \in G\), so \(\, B(z; r) \subset G_1 \implies G_1\,\) is open. Likewise \(G_2\) is open.
\(G_1\cup G_2 = G\quad G_1, G_2\) are open and \(\, G_1\cap G_2 = \emptyset \implies \) either \(G_1 = \emptyset \) or \(G_2 = \emptyset \). We know that \(\, G_1 \neq \emptyset \) so \(G_2 = \emptyset \,\) i. e \(\, G_1 = G\). □
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