Results 1 to 11 of 11

Thread: Open set, closed set or neither

  1. #1
    Newbie
    Joined
    Jan 2017
    From
    Netherlands
    Posts
    9

    Open set, closed set or neither

    I want to check for this set to be open, closed or neither

    I = {x ∈ R3 : 1 ≤ x1 ≤ 3,0 ≤ x2,−1 ≥ x3}

    I think it is closed, but I don't know how to show it.
    I have already shown that it is not an open set by saying:
    you cannot say that there is for any point x in I an open ball with center z and radius r. Because i we take x1 = 3, then we cannot find a r such that ||x-z||<r.
    I know that you can prove that a set is closed by proving that the complement of the set is open, but if I take R3\I, how can I show that for every x there is a open ball? Or have I do it another way?

    Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,606
    Thanks
    2368

    Re: Open set, closed set or neither

    both x2 and x3 are infinite intervals

    how is I going to be closed?
    Last edited by romsek; Jan 30th 2017 at 05:56 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2017
    From
    Netherlands
    Posts
    9

    Re: Open set, closed set or neither

    But how can you prove it that is not open nor closed?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,606
    Thanks
    2368

    Re: Open set, closed set or neither

    Quote Originally Posted by Cyn1 View Post
    But how can you prove it that is not open nor closed?
    Ignore what I wrote. It's incorrect. Intervals of the sort $[a,\infty)$ are closed.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,606
    Thanks
    2368

    Re: Open set, closed set or neither

    the complement is

    $\left((-\infty,1)\cup (3, \infty)\right) \times (-\infty, 0) \times (-1, \infty) = (-\infty,1) \times (-\infty, 0) \times (-1, \infty) \bigcup (3, \infty) \times (-\infty, 0) \times (-1, \infty)$

    intervals of the form $(-\infty, a)$ or $(a, \infty)$ are open as they contain none of their endpoints.

    so $I^c$ is the union of two open 3D intervals which is of course open.

    Since $I^c$ is open $I$ is closed

    What you need to show is the assertion about open semi-infinite intervals being open which isn't much different from showing finite open intervals are open.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jan 2017
    From
    Netherlands
    Posts
    9

    Re: Open set, closed set or neither

    Thank you,

    But I have still don't understand it totally.
    Have the points in this set satisfy all of the conditions? I mean that the point (2,2,-5) is a point in this set, but is (2,2,2) also a point in this set?
    And how can I show that the complement isn't open either? The complement is I^c= {x ∈ R3 : 1>x1, x1>3, 0 > x2, −1 < x3}.

    Can someone help me?
    Thank you
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jan 2017
    From
    Netherlands
    Posts
    9

    Re: Open set, closed set or neither

    I am sorry, I saw you answer just a few minutes ago. But I have still one question: Have the points in this set satisfy all of the conditions? I mean that the point (2,2,-5) is a point in this set, but is (2,2,2) also a point in this set?
    Thank you
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,606
    Thanks
    2368

    Re: Open set, closed set or neither

    Quote Originally Posted by Cyn1 View Post
    I am sorry, I saw you answer just a few minutes ago. But I have still one question: Have the points in this set satisfy all of the conditions? I mean that the point (2,2,-5) is a point in this set, but is (2,2,2) also a point in this set?
    Thank you
    $2 > -1$ so $(2,2,2) \not \in I$

    I'm not really sure what's important about that.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,198
    Thanks
    2626
    Awards
    1

    Re: Open set, closed set or neither

    Quote Originally Posted by Cyn1 View Post
    I want to check for this set to be open, closed or neither
    I = {x ∈ R3 : 1 ≤ x1 ≤ 3,0 ≤ x2,−1 ≥ x3}
    I think it is closed, but I don't know how to show it.
    I have already shown that it is not an open set by saying:
    you cannot say that there is for any point x in I an open ball with center z and radius r. Because i we take x1 = 3, then we cannot find a r such that ||x-z||<r.
    I know that you can prove that a set is closed by proving that the complement of the set is open, but if I take R3\I, how can I show that for every x there is a open ball? Or have I do it another way?
    If $\mathcal{O}$ is an open set then $\mathcal{O}$ cannot contain any of its own boundary points.
    Is it true that $(3,1,-1)\in\mathcal{I}~?$

    Any closed set, $\mathcal{M}$ must contain contain all of its limit points.

    Those two facts are very useful in answering this sort of question. The fact that the set $\mathcal{I}$ is connected makes this somewhat easier.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Jan 2017
    From
    Netherlands
    Posts
    9

    Re: Open set, closed set or neither

    Thank you,

    But how can you show that a set M contains all of its limit points?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Jan 2017
    From
    Netherlands
    Posts
    9

    Re: Open set, closed set or neither

    But actually, I have to prove that the complement of I is open and so that I is closed. But how can you prove that it holds for every x?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Open or Closed?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Jun 22nd 2011, 01:32 PM
  2. Closed or open set
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Apr 4th 2011, 06:30 PM
  3. Set in R^2 which is neither open nor closed.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 3rd 2010, 11:46 PM
  4. open and closed
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 26th 2010, 09:26 PM
  5. open? closed? both?
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Apr 10th 2010, 09:56 AM

Search Tags


/mathhelpforum @mathhelpforum