My professor told me that the real line is connected in any topology. But i am thinking that if for example we consider the discrete topology in $\mathbf R$ i.e every subset is open then for any $x$ in $\mathbf R$ $(-\infty,x) \cup [x,\infty)$ would be a partition of open sets such that their union is $\mathbf R$ and thus $\mathbf R$ is disconnected in discrete topology. Is what i am saying wrong? why?
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6Any space with more than one point is disconnected in the discrete topology. You're right. – Daniel Fischer Feb 05 '14 at 18:38
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Cheers that makes sense. I guess my professor got confused... – lukanikos Feb 05 '14 at 18:40
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2Perhaps you misunderstood your professor. Or perhaps he misspoke. In either case, ask him, not us. – GEdgar Feb 05 '14 at 18:42
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1@GEdgar I just want to clarify whether this is something I'd like to take to meta: what part of OP's question do you find unsuitable to this site? – Jonathan Y. Feb 05 '14 at 19:01
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As a comment, I gave the OP some advice. Sometimes I do that even when the question is suitable for this site... – GEdgar Feb 05 '14 at 21:58
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You are correct. The only discrete spaces that are connected are those with at most one point, since otherwise, they have a non-empty proper subset that is both open and closed.
Cameron Buie
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