Results 1 to 3 of 3

Math Help - Limit point proof

  1. #1
    Newbie
    Joined
    Oct 2007
    Posts
    5

    Limit point proof

    I am having a hard time with a proof:

    Prove that the set of limit points of a set is closed.

    At first it seemed easy and from a logical standpoint it makes sense but I am not sure where to start.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by jlindbo View Post
    I am having a hard time with a proof:

    Prove that the set of limit points of a set is closed.

    At first it seemed easy and from a logical standpoint it makes sense but I am not sure where to start.
    what definition are you using for a "limit point"? a "closed set"?

    strategies you may want to explore:

    - show that the compliment of the set of limit points is open (this is perhaps the easiest way)
    - show that if S is the set of limit points, it will contain all the limits of its convergent sequences (consisting of points in S)
    - recall also that x is a limit point of a set S if it lies in the closure of S - {x}. And show that this set is closed. (this amounts to showing that the closure of a set is closed, i think)
    - you can also show that for any point in S (the set of limit points of a set) that has the property that any open ball centered at that point intersects S and S compliment (that is, it is a boundary point of S) will be contained in S. (we know that such points exists based on one of the definitions for a limit point)*. thus we would show that S contains all its boundary points and is hence closed

    *) Definition: Let S be a subset of a topological space X. We say that a point x in X is a limit point of S if every open set containing x also contains a point of S other than x itself.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    I think that "limit point" here means the let of all accumulation points.

    Here
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: October 18th 2011, 05:48 AM
  2. Limit point compactness
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: September 28th 2011, 07:38 PM
  3. Sequence and limit point
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 20th 2010, 02:45 AM
  4. Limit Point Proof.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 5th 2009, 08:56 PM
  5. uncountable set: limit point
    Posted in the Calculus Forum
    Replies: 13
    Last Post: July 10th 2008, 07:53 PM

Search Tags


/mathhelpforum @mathhelpforum