Results 1 to 6 of 6

Math Help - The interior of a set is open and the boundary and closure of a set is closed

  1. #1
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419

    The interior of a set is open and the boundary and closure of a set is closed

    So I am pretty sure my proof is correct, just want to verify for correctness and rigor.

    Let a\in int(S). Then there exists \epsilon > 0 such that B(\epsilon , a) \subset S. Now suppose x\in B(\epsilon , a). Since B(\epsilon, a) is open, there exists \delta > 0 such that B(\delta , x) \subset B(\epsilon , a) \subset S; in other words, for any x\in B(\epsilon , a) there is a ball centered at x contained solely in S, which means x\in int(S). So given a\in int(S) there exists r > 0, namely r = \epsilon, such that B(r,a) \subset int(S), so int(S) is open.

    So now consider \partial S. Then (\partial S)^{c} = int(S) \cup int(S^{c}) since if x is not a point such that every ball centered at x intersects S and S^{c}, then there exists some ball centered at x that is either contained solely in S or S^{c}. So we have that (\partial S)^{c} is the union of two open sets, which is open, so \partial S is closed. Similarly, if we consider cl(S), then (cl(S))^{c} = S^{c}\cap(int(S)\cup int(S^{c})) = int(S^{c}), which is open, so cl(S) is closed.

    Q.E.D.

    So I think this is a perfectly valid and rigorous proof, but any feedback would be appreciated (I am somewhat unsure on the closed portions of the proof, especially the claims about what the complements of the boundary of S and the closure of S actually equal). Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    I find your proof very hard to follow. That is not saying it is incorrect!
    I think it is too complicated.

    It is easy to prove that any open set is simply the union of balls.
    The interior is just the union of balls in it.
    The complement of the closure is just the union of balls in it.
    The complement of the boundary is just the union of balls in it.

    To follow that last bit, think this way.
    If t\notin \beta(A) the there is a ball \mathcal{B}(t;\delta) that is a subset of \mathcal{I}(A) or the complement of A
    That is it contains only points of A or points not in A.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419
    Hm, thanks. I guess I made it more complicated if not for the fact that this is my first time learning about open/closed sets in terms of interior points and balls, so when it asks if a set is open I go back to the basic definition that any point in an open set has some ball centered at that point contained solely in the set. (At least that is how the textbook I am using defines it). And since this is a new area of mathematical study for me, I don't know how to actually show that an open set is the union of balls.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Actually that is an axiom.
    The union of any set of open sets is itself an open set.
    That should have been stated up front.
    Then by definition, any ball is an open set.
    So any union of balls is open, by the axiom.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419
    Ah, okay. I guess my textbook (Advanced Calculus by Gerald B. Folland) does things differently. Thanks again.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Aug 2010
    Posts
    44
    The interior of a set M is the largest open set, which is still a subset of M or equivalently the union of all open subsets of M.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Interior, Closure and Boundary of sets
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 18th 2011, 11:48 AM
  2. Closed, open, int, adh and boundary
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 31st 2011, 04:37 AM
  3. Interior, closure, open set in topological space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 2nd 2011, 06:46 PM
  4. Interior, Boundary, and Closure of Sets
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 22nd 2009, 04:01 PM
  5. open/closed/interior
    Posted in the Differential Geometry Forum
    Replies: 12
    Last Post: May 7th 2009, 12:21 AM

Search Tags


/mathhelpforum @mathhelpforum