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Math Help - Open sets and sets of interior points

  1. #1
    Senior Member I-Think's Avatar
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    Open sets and sets of interior points

    Need verification of a proof please

    Question
    Let  E^o be the set of all interior points of a set E.
    Prove E^o is always open

    Proof
    Consider p\in{E^o}. \exists a neighborhood N of p such that N\subset{E}

    This N is an open set, so \forall{q\in{N}}, \exists a neighborhood M of q so M\subset{N\subset{E}}
    So N consists of interior points of E, so E^o is open.
    QED

    Proof correct? Any alternatives or ways to make it cleaner/more elegant?
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  2. #2
    MHF Contributor

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    Re: Open sets and sets of interior points

    Quote Originally Posted by I-Think View Post
    Question
    Let  E^o be the set of all interior points of a set E.
    Prove E^o is always open
    Proof
    Consider p\in{E^o}. \exists a neighborhood N of p such that N\subset{E}
    This N is an open set, so \forall{q\in{N}}, \exists a neighborhood M of q so M\subset{N\subset{E}}
    So N consists of interior points of E, so E^o is open.
    QED
    It is correct. Basically you are noting that a neighborhood is a neighborhood of each of its points.
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