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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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This is an interesting question, but the way it's phrased makes it seem like a homework problem. – Grumpy Parsnip Aug 23 '11 at 12:18
Let me assure you that it is homework! I'm older than posting my homework here. – Ehsan M. Kermani Aug 23 '11 at 12:28
up vote 3 down vote accepted

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....... – Andrea Mori Aug 23 '11 at 13:44

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