We know that a subspace of a connected space can be disconnected eg. $\mathbf{Q} \in \mathbf{R}$ where $\mathbf{R}$ is connected but $\mathbf{Q}$ is totally disconnected as a subspace. My question is, "Does there exists a topological space such that every proper subspace (except singletons) is disconnected but the whole space is connected".
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1Singletons are always connected. You surely mean every non-trivial subspace with at least two elements? – Stefan Hamcke Apr 19 '14 at 13:30
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yes otherwise it would be trivial – user2902293 Apr 19 '14 at 13:35
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1I don't know about every proper subspace, but for an example of a connected space which gets totally disconnected after you remove one point see Knaster-Kuratowski fan – user2345215 Apr 19 '14 at 13:47
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3The two-point indiscrete (trivial) space? (This is a bit of a degenerate example, I know.) – user642796 Apr 19 '14 at 13:55
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2Niels Diepeveen's answer at http://math.stackexchange.com/questions/634787/is-there-an-infinite-connected-topological-space-such-that-every-space-obtained should answer your question since such a space would become totally disconnected upon removing any point. – Stefan Hamcke Apr 19 '14 at 14:04