Results 1 to 2 of 2

Math Help - [SOLVED] Disconnected Set

  1. #1
    Member
    Joined
    Jul 2008
    Posts
    119

    [SOLVED] Disconnected Set

    Problem:
    Suppose that A is a subset of \mathbb{R}^n that fails to be connected(i.e. disconnected), and let U, V be open subset of \mathbb{R}^n that separate A.
    Suppose that B is a subset of A that is connected. Prove that either  B \subseteq U or  B \subseteq V

    Proof:
    Let U, V be disjoint open subsets that separate A, then by definition we know three things:
    (i)  A \cap U = \emptyset and  A \cap V = \emptyset
    (ii)  (A \cap U ) \cap (A \cap V) = \emptyset
    (iii)  (A \cup U ) \cap ( A \cup V) = A

    Let  B \subset A that is connected.

    This is what I believe it to be, but don't understand why it's true.
     B \cap U \neq \emptyset \implies B \subseteq U
     B \cap V \neq \emptyset \implies B \subseteq V

    If B is a subset of A, then shouldn't U and V disconnect B as well? Thus, B can not be contained in U and B can not be contained in V.

    Thank you so much for reading. Any help is greatly appreciated.
    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 Paperwings View Post
    Problem:
    Suppose that A is a subset of \mathbb{R}^n that fails to be connected(i.e. disconnected), and let U, V be open subset of \mathbb{R}^n that separate A.
    Suppose that B is a subset of A that is connected. Prove that either  B \subseteq U or  B \subseteq V
    Try this.
    If B\subseteq A then (B \cap U) \cup (B\cap V) = B.
    Now (B\cap U)\cap (B\cap V) = \emptyset.
    Therefore B\cap U = B or B\cap V = B.
    Thus, B\subseteq U or B\subseteq V.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Disconnected Graphs
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: March 8th 2012, 06:16 AM
  2. Prove that E is disconnected
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: December 1st 2011, 08:44 AM
  3. Totally Disconnected Compact Set with Positive Measure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: December 28th 2010, 10:26 AM
  4. If M is disconnected....
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 6th 2010, 12:51 PM
  5. Proof of disconnected sets
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 14th 2008, 03:58 PM

Search Tags


/mathhelpforum @mathhelpforum