Results 1 to 3 of 3

Math Help - Topology: Closure

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    13

    Topology: Closure

    For sets A and B in a topological space X, the following hold:
    (1) Cl(X-A)=X-Int(A)
    (2) Int(A)∩ Int(B) =Int (A∩B)
    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
    For sets A and B in a topological space X, the following hold:
    (2) Int(A)∩ Int(B) =Int (A∩B)
    If x\in \text{int}(A) \cap \text{int}(B) then x\in \text{int}(A) \text{ and }x\in \text{int}(B).
    This means there are open subsets U,V so x\in U\subseteq A \text{ and }x\in V\subseteq B.
    Thus, x\in U\cap V \subseteq A\cap B, remember that U\cap V is open subset.
    Thus, x\in \text{int}(A\cap B).

    If x\in \text{int}(A\cap B) then x\in U\subseteq A\cap B for some open subset U.
    But then x\in U\subseteq A \text{ and }x\in U\subseteq B.
    Therefore, x\in \text{int}(A) \text{ and }x\in \text{int}(B) \implies x\in \text{int}(A)\cap \text{int}(B).

    Thus, \text{int}(A\cap B) = \text{int}(A)\cap \text{int}(B).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by horowitz View Post
    (1) Cl(X-A)=X-Int(A)
    Let x\in \overline{X-A} then it means for any open set U with x\in U we have that U\cap (X-A)\not = \emptyset. But then there is not open set U with x\in U so that U\subseteq A (otherwise the intersection would be empty). Therefore, x is not and interior point of A, thus, x\in X - \text{int}(A). Conversely, let x\in X - \text{int}(A), this means x\in X and x\not \in \text{int}(A). But if x\not \in \text{int}(A) it means for any open set U so that x\in U we have that U\not  \subseteq A. Thus, for any open set U with x\in U we see that U\cap X - A\not = \emptyset which means x lies on the closure of X-A.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. find the closure of A in the co-finite topology
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 13th 2011, 11:55 AM
  2. Replies: 6
    Last Post: March 28th 2010, 01:10 PM
  3. closure of sets (topology)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 17th 2009, 06:38 AM
  4. Finding the closure of sets in a given topology
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 7th 2009, 10:01 AM
  5. some topology questions about closure and boundary
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 16th 2008, 05:05 PM

Search Tags


/mathhelpforum @mathhelpforum