Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By SlipEternal

Math Help - Why is the empty subset of a metric space always connected ?

  1. #1
    Member
    Joined
    Sep 2012
    From
    india
    Posts
    78

    Question Why is the empty subset of a metric space always connected ?

    How can i show that the empty subset of a metric space,X always connected ?

    It is empty and so will have and so will have an empty boundary.That doesn't seem to be enough.

    Also, my book says that, no other finite set can be connected,I don't really understand this, because every finite set will be a non empty proper subset of X and will always be closed (and never both open and closed) so shouldn't it be connected ?

    Any help will be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,937
    Thanks
    785

    Re: Why is the empty subset of a metric space always connected ?

    The empty subset trivially satisfies the conditions of connectedness. Can you represent the empty subset as the disjoint union of two nonempty open sets?

    Let x \in X. Then \{x\} is also connected (I don't know why your book would not include singleton sets. They are also connected). Again, it cannot be represented by the disjoint union of two or more nonempty open sets.

    Let A \subseteq X with 2 \le \text{card}(A) \in \mathbb{N}. Let D = \{d(a,b) \mid a,b \in A, a\neq b\}. D must be a finite set since A is. So, let d = \min(D). Then for any a \in A, B\left(a;\dfrac{d}{2}\right) = \{a\} is an open subset of A. Since every set containing a single point is an open subset of A, if A has more than one point, let U = \{a\} and V = A \setminus \{a\}. By induction, you can show that V is open and nonempty. And A = U \cup V.
    Thanks from mrmaaza123
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Metric space subset
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 2nd 2012, 01:46 PM
  2. Replies: 1
    Last Post: September 26th 2012, 10:06 AM
  3. Replies: 1
    Last Post: March 11th 2012, 09:08 AM
  4. Every path-connected metric space is connected
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 13th 2011, 07:31 AM
  5. A finite subset of a metric space
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 10th 2009, 06:48 PM

Search Tags


/mathhelpforum @mathhelpforum