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Math Help - Is the Excluded Point Topology a scattered space?

  1. #1
    Super Member Matt Westwood's Avatar
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    Is the Excluded Point Topology a scattered space?

    Let S be a set.

    Let p \in S.

    Then the excluded point topology T = (S, \tau) is given by:

    A subset of  H \in \tau is open iff p \notin H .


    The argument given in Counterexamples in Topology by Steen and Seebach (counterexamples 13-15: 5) goes:

    Let x \in S. Then \{x\} \subseteq S is open in T as  \notin \{x\} .

    So by definition x can not be a limit point and so is an isolated point.

    So any subset H \subseteq S is not dense-in-itself, because it contains isolated points.

    So by definition T is scattered.


    There's a flaw in the above, in that p is easily shown to be a limit point (the only one possible in T, in fact) and so \{p\} has no isolated points in it, so is dense-in-itself.

    So it appears that T is not scattered after all, as it contains (exactly) one dense-in-itself subset.


    Is Steen and Seebach wrong? Or have I missed something, e.g. " H is required to be open to be described as dense-in-itself" or something? But I can't find any such words.
    Last edited by Plato; June 8th 2011 at 02:21 PM. Reason: de-latexed as the commands don't work
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  2. #2
    MHF Contributor

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    Please use the [tex][/tex] tags.
    [tex]\{x\}\subseteq S[/tex] gives  \{x\}\subseteq S.
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