Complex Analysis: Show the union of 2 regions is connected [duplicate]

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Let G1 and G2 be two regions. Suppose that G1 ∩ G2 ≠ 0. Show that G1 U G2 is connected.

I know I have to first show that G1 U G2 is open and then show it is connected but I have no idea where to start or how to even prove this.

marked as duplicate by José Carlos Santos, Namaste, Xander Henderson, Lord Shark the Unknown, Deepesh MeenaSep 25 '18 at 4:58

Let $$G_1$$ and $$G_2$$ be connected subspaces of $$\mathbb{C}$$. We want to show that $$G:=G_1 \cup G_2$$ is connected. Let $$A$$ and $$B$$ be two disjoint open subsets of $$G$$ such that $$G = A \cup B$$. Since $$G_1$$ is connected, we must have either $$G_1 \subseteq A$$ or $$G_1 \subseteq B$$. Without loss of generality, assume $$G_1 \subseteq A$$. Since $$G_2$$ is also connected, we either have $$G_2 \subseteq A$$ or $$G_2 \subseteq B$$. Let $$z \in G_1 \cap G_2$$ and observe that $$z \in G_1 \subseteq A$$. Because $$A \cap B = \varnothing$$, the only possibility is $$G_2 \subseteq A$$.
It follows that $$G_1 \cup G_2 \subseteq A \subseteq G = G_1 \cup G_2$$. We conclude that $$A = G_1 \cup G_2$$ and that $$B = \varnothing$$. In other words, $$G_1 \cup G_2$$ does not admit a separation and must be connected.
To show that $$G_1 \cup G_2$$ is open, simply use that the union of open sets is again open.