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Math Help - how to show dense in Finite complement topology

  1. #1
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    how to show dense in Finite complement topology

    Let U be any open set in the finite complement topology on .

    Show that U is dense in
    Last edited by Jhevon; April 26th 2009 at 12:59 PM. Reason: restored post
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  2. #2
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    Hello,
    Quote Originally Posted by monkey.brains View Post
    Let U be any open set in the finite complement topology on .

    Show that U is dense in
    U is an open set in this topology. Let's assume that U is not the empty set.
    Hence \mathbb{R}\backslash U is finite and is a closed set of this topology.
    Similarly, it is obvious that any closed set of this topology is finite, except \mathbb{R}, which is closed and open.

    We have U \cup \{\mathbb{R}\backslash U\}=\mathbb{R}
    But \mathbb{R} is an infinite set, and \mathbb{R}\backslash U is a finite set.
    Thus U is an infinite set.

    A definition of U being dense in \mathbb{R}, is that the unique closed set containing U is \mathbb{R} itself.

    But a closed set in the finite complement topology is, as said before, finite.
    Thus there is no possible closed set containing U, except \mathbb{R}
    This proves that U is dense in \mathbb{R}



    Now you have to study the situation where U= \emptyset. It looks simple, but I'm quite confused... can you try it ?
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