# 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.