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Let $X$ be a compact connected Hausdorff space with more than one point. Prove that there is point $x \in X$ s.t. $X \setminus \{x\}$ is connected.

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    Let me assure you that it is homework! I'm older than posting my homework here.2011-08-23

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I was typing up an answer, but I must go. So I will refer you to the answer. In this paper, at the bottom of page 380, there is a proof that there are at least 2 non-cut points.

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    You had a nice proof which was unfortunately too long for your *time* margin to contain.......2011-08-23