Is it possible to construct a connected subset of the plane with the property that removal of any single point makes it totally disconnected? Any answer is appreciated..Thanks!!
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I suppose that a singleton set might satisfy the definition trivially. After removal, the resulting set has the property that any two points in it can be separated... – Mark McClure Oct 16 '14 at 19:32
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Yes, the Knaster-Kuratowski fan is such an example. – Hayden Oct 16 '14 at 19:42
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@Hayden The Knaster-Kuratowski fan only becomes totally disconnected if we remove a certain point $p$, not if we remove any point. – Dietrich Burde Oct 16 '14 at 19:47
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Google "explosion point" along with "connected". – Dave L. Renfro Oct 16 '14 at 19:55
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@DietrichBurde Good point, the OP did say "any". – Hayden Oct 16 '14 at 20:12
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No, it is impossible, if we exclude the trivial case of a singleton. Such a point is called a dispersion point. A space can only have one dispersion point. So we cannot remove any single point to obtain a totally disconnected space.
However, if we only want that the removal of one single point makes the space totally disconnected, then the Knaster-Kuratowski fan, or Cantor's teepee is a well-known example.
Dietrich Burde
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Dear all, thanks for the comments. I would assume that we are all interested in NON-TRIVIAL examples. So the question can be corrected as: Can one find a non-trivial connected subset $A$ of the plane such that $A{x}$ is disconnected for every $x\in A$. So what we are looking for is a non-tivial subset of the plane with the property that Whatever point you remove from the set, it becomes totally disconnected. I'm sorry for any misleading..Thanks again! – Aneesh M Oct 17 '14 at 20:19
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Dear Aneesh, there is no non-trivial subset of the plane with this property - see my answer. – Dietrich Burde Oct 18 '14 at 18:11