Results 1 to 4 of 4

Math Help - Metric Spaces and Topology

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    1

    Red face Metric Spaces and Topology

    A and B are any sets in a metric space (X, d)
    Prove that
    (A U B )' = A' U B'

    ' meaning interior.

    And then prove that

    (A U B ) closure = A closure U B closure


    I know that you have to prove it both ways, and that the proofs will be relativley similar, and I know that they are true. I'm just really confused on where to get started
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by CollegeMathKid View Post
    A and B are any sets in a metric space (X, d)
    Prove that
    (A U B )' = A' U B'

    ' meaning interior.
    This is not true: Take X= \mathbb{R}, A=(0,1) \cap \mathbb{Q} B= (0,1) \cap (\mathbb{R} - \mathbb{Q} ) then int(A)=int(B)=\emptyset but int(A \cup B)=(0,1)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1
    Quote Originally Posted by CollegeMathKid View Post
    A and B are any sets in a metric space (X, d)
    Prove that
    (A U B )' = A' U B', ' meaning interior.
    That is false.
    Let A=[1,2]~\&~B=[2,3] then 2\in(A\cup B)^o=(1,3).

    BUT 2\not\in A^o\cup B^o=(1,2)\cup(2,3).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    However, you should be able to prove that int(A)\cup int(B)\subseteq int(A\cup B).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 15th 2011, 03:00 PM
  2. Metric Topology
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 28th 2010, 07:30 AM
  3. topology/metric spaces questions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 26th 2008, 03:38 AM
  4. Metric Spaces/Topology help
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: November 25th 2008, 11:49 PM
  5. Replies: 1
    Last Post: April 8th 2008, 07:00 PM

Search Tags


/mathhelpforum @mathhelpforum