Let $\{A_i\}_{i \in I}$ be a family of connected subsets of a metric space $X$ ($I$ is some set of indices). Show that if the intersection $\bigcap A_i \neq \emptyset$ , then $\bigcup A_i$ is connected.
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Sorry. Don't know how to write math characters on here. – user2684794 Nov 03 '15 at 23:52
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There’s a tutorial and quick reference here for writing mathematics on this site. – Brian M. Scott Nov 04 '15 at 01:23
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Since $\bigcap A_i \ne \emptyset$, let $x \in \bigcap A_i$. Now if $U$ and $V$ are non-empty disjoint open sets separating $\bigcup A_i$ then $x \in U$ or $x \in V$. Without loss of generality, suppose $x \in U$. Now, where does $x$ come from and what can we say about it?
Kevin Sheng
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I guess you could say x comes from the intersection of A_i but is not an element of V – user2684794 Nov 04 '15 at 00:25
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