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Math Help - If M is disconnected....

  1. #1
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    If M is disconnected....

    Let (M,d) be a metric space. Prove the equivalence of the following:
    (i)M is disconnected
    (ii)There exists two nonempty, disjoint closed sets in M whose union is M
    (iii)There exists two nonempty, disjoint open sets in M whose union is M
    (iv)There exists a set A in M such that emptyset != A != M, and A is both open and closed.

    these are basically the definitions of a disconnected set aren't they? I have no idea how to prove these =/
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by cp05 View Post
    Let (M,d) be a metric space. Prove the equivalence of the following:
    (i)M is disconnected
    (ii)There exists two nonempty, disjoint closed sets in M whose union is M
    (iii)There exists two nonempty, disjoint open sets in M whose union is M
    (iv)There exists a set A in M such that emptyset != A != M, and A is both open and closed.

    these are basically the definitions of a disconnected set aren't they? I have no idea how to prove these =/
    Well what's your definition of disconnected? I assume that X can be written as the disjoint union of open sets.

    i)\implies ii): This is clear since if A\amalg B=X (where A,B are open) then A=X-B and B=X-A both of which are disjoint and closed. So then, X=A\amalg B=\left(X-B\right)\amalg \left(X-A\right)

    iii)\implies IV) Suppose that A\amalg B=X is a disconnection of X (with A,B closed). Then, B is closed by assumption and B=X-A and since A is closed it folllows that B is open.



    Oops...I just noticed that I supposed wrong your definition of disconnected. What is it?
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  3. #3
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    Disconnected if there exists sets A and B in M such that
    1. A != null != B
    2. D= A U B
    3. clos(A) n B=null=A n clos(B) iff A n B=null
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by cp05 View Post
    Disconnected if there exists sets A and B in M such that
    1. A != null != B
    2. D= A U B
    3. clos(A) n B=null=A n clos(B) iff A n B=null
    Oh god, separated sets. That's so gross. I don't understand the last part though.

    Usually it is " X=A\cup B where A,B are non-empty and \overline{A}\cap B=B\cap\overline{A}=\varnothing"

    Is that what you mean to say?
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  5. #5
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    yes
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by cp05 View Post
    yes
    Look here.
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