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Math Help - Ultrafilters Topology

  1. #1
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    Ultrafilters Topology

    Let X be a topological space. let X' be the collection of all the ultrafilters on X.
    we say that a set S\subseteq X' is open and belongs to the base of X' if

    there is an open U\subseteq X, such that U'\in S \Leftrightarrow U\in U'.

    Let N' be the space of ultrafilters (filters on N with the discrete topology) as explained above.
    Prove that if a sequence U_{i \ i\in N} converge in N' \  \ U_i=U_j for every large enough i,j.

    any help is appreciated...
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by aharonidan View Post
    Let X be a topological space. let X' be the collection of all the ultrafilters on X.
    we say that a set S\subseteq X' is open and belongs to the base of X' if

    there is an open U\subseteq X, such that U'\in S \Leftrightarrow U\in U'.

    Let N' be the space of ultrafilters (filters on N with the discrete topology) as explained above.
    Prove that if a sequence U_{i \ i\in N} converge in N' \  \ U_i=U_j for every large enough i,j.

    any help is appreciated...
    Maybe I'm being deceived, but isn't this just a consequence that for any discrete space X the only convergent sequences are eventually constant?
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  3. #3
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    Opalg's Avatar
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    Quote Originally Posted by aharonidan View Post
    Let X be a topological space. let X' be the collection of all the ultrafilters on X.
    we say that a set S\subseteq X' is open and belongs to the base of X' if

    there is an open U\subseteq X, such that U'\in S \Leftrightarrow U\in U'.

    Let N' be the space of ultrafilters (filters on N with the discrete topology) as explained above.
    Prove that if a sequence U_{i \ i\in N} converge in N' \  \ U_i=U_j for every large enough i,j.

    any help is appreciated...
    Another way to phrase this problem is to say that every convergent sequence in the Stone–Čech compactification of N is eventually constant. I believe that this is a standard result in the theory of the Stone–Čech compactification, but I do not know of a proof. You might try looking here for ideas and further references.
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