Results 1 to 4 of 4

Math Help - proving a set is closed

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    42

    proving a set is closed

    how can I prove that [a, b] is a closed set?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2010
    Posts
    470
    A set is closed if its complement is open.

    In this case prove:
    (-inf, a) union (b, +inf)
    is open.

    To prove this set is open just use the definition.
    Intuitively: show for every point in the open set, there is a small neighborhood around that point that is also in the set.

    ====

    You can also use the lemma that a set is closed iff the closure is the set itself. I.e. show the closure of [a,b] is just [a,b].
    Last edited by snowtea; December 14th 2010 at 10:12 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by cottekr View Post
    how can I prove that [a, b] is a closed set?
    It depends what your definition of closed is!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Jul 2009
    Posts
    107
    Quote Originally Posted by cottekr View Post
    how can I prove that [a, b] is a closed set?

    To prove that [a,b] is closed you must prove:

    1) For all x>b there exists an ε>0 ,such that:

    ......For all ,y : |x-y|<\epsilon\Longrightarrow y>b.

    2) For all x<a there exists an ε>0 , such that:

    ......For all ,y : |x-y|<\epsilon\Longrightarrow y<a
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving a set is closed and convex.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 7th 2010, 08:32 PM
  2. Proving a set is closed
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: November 17th 2009, 07:33 AM
  3. Proving a set is closed
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: November 16th 2009, 12:42 PM
  4. Proving a set closed
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 24th 2009, 11:15 AM
  5. Proving the 'Closed Ball' is closed
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 13th 2009, 06:02 PM

Search Tags


/mathhelpforum @mathhelpforum