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Math Help - Finite Compliment Topology

  1. #1
    Forum Admin topsquark's Avatar
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    Finite Compliment Topology

    This should probably be an easy one.

    The "finite compliment topology" is defined by my text as the collection of sets U of a set X such that X - U is either finite or all of X.

    Nothing in the definition says that X must be finite, but I was looking at an example of a finite set and it struck me that the FCT on a finite set X is the same as the discrete topology on that same X. (The discrete topology is defined as the collection of all subsets of X.) I can verify this for a given set X but I am at a loss of how to prove it in general. I've been toying with some kind of construction or an induction proof, but can't quite seem to get a good start on it. Any thoughts?

    Thanks
    -Dan
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by topsquark View Post
    This should probably be an easy one.

    The "finite compliment topology" is defined by my text as the collection of sets U of a set X such that X - U is either finite or all of X.

    Nothing in the definition says that X must be finite, but I was looking at an example of a finite set and it struck me that the FCT on a finite set X is the same as the discrete topology on that same X. (The discrete topology is defined as the collection of all subsets of X.) I can verify this for a given set X but I am at a loss of how to prove it in general. I've been toying with some kind of construction or an induction proof, but can't quite seem to get a good start on it. Any thoughts?

    Thanks
    -Dan
    Let U\subseteq X where \#(X)<\infty then \#\left(X-U\right)\leqslant \#(X)<\infty and so U is open. Thus, the topology is just the power set, which by definition means that it's the discrete topology.


    Remark: In fact, the reason that the cofinite topology is important is it is the coarsest T_1 topology on any set (up to homeomorphism). That said, more generally one can show that any finite T_1 space is discrete.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Let U\subseteq X where \#(X)<\infty then \#\left(X-U\right)\leqslant \#(X)<\infty and so U is open. Thus, the topology is just the power set, which by definition means that it's the discrete topology.


    Remark: In fact, the reason that the cofinite topology is important is it is the coarsest T_1 topology on any set (up to homeomorphism). That said, more generally one can show that any finite T_1 space is discrete.
    Very concise! Thank you.

    -Dan
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