Results 1 to 2 of 2

Thread: Open sets and sets of interior points

  1. #1
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288

    Open sets and sets of interior points

    Need verification of a proof please

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

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

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

    Proof correct? Any alternatives or ways to make it cleaner/more elegant?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,736
    Thanks
    2811
    Awards
    1

    Re: Open sets and sets of interior points

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

Similar Math Help Forum Discussions

  1. Metric spaces, open sets, and closed sets
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Mar 16th 2011, 05:17 PM
  2. Open sets and closed sets
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 30th 2010, 10:05 AM
  3. Open sets and closed sets
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 11th 2010, 05:57 PM
  4. Proof: Interior points and closed sets.
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Feb 1st 2010, 09:30 AM
  5. Open sets and interior points Complex
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 6th 2009, 03:26 PM

Search Tags


/mathhelpforum @mathhelpforum