Results 1 to 2 of 2

Math Help - Closure

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    13

    Closure

    Let X be a topological space, A be a subset of X, and y be an element of X. Then y ∈ Cl(A) if and only if every open set containing y intersects A.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by horowitz View Post
    Let X be a topological space, A be a subset of X, and y be an element of X. Then y ∈ Cl(A) if and only if every open set containing y intersects A.
    A point y is on the boundary of A iff for any for any open set U containing y we have that U intersects both A and X-A. If y is on the closure of A then y is in A or the boundary of A. If y is in A then every open set containing y obviously intersects A. And if y is on the boundary of A then any open set containing y intersects A (by what was said above). Thus, any open set containing y intersects A. Try doing the backwards direction.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Relation between topological closure and algebraic closure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 4th 2010, 02:45 PM
  2. closure
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: March 27th 2010, 12:01 AM
  3. Closure
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 19th 2009, 01:59 AM
  4. closure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 27th 2009, 01:03 PM
  5. Replies: 6
    Last Post: February 11th 2009, 12:56 PM

Search Tags


/mathhelpforum @mathhelpforum