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Connected Spaces

  1. Apr 9, 2010 #1
    Can a disconnected space be a disjoint union of two infinite sets?
    Must the disjoint subspaces be finite?
     
  2. jcsd
  3. Apr 9, 2010 #2

    dx

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    No, the connected components need not be finite, nor do they have to be compact.
     
  4. Apr 9, 2010 #3

    Landau

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    Sure. E.g.[tex]T=[0,1]\cup [2,3]\subset\mathbb{R}[/tex].

    Well, just take any two disjoint infinite open subsets of some topological space, then their union is by definition disconnected.
     
  5. Apr 10, 2010 #4
    Landau wrote, in part:

    " Well, just take any two disjoint infinite open subsets of some topological space, then their union is by definition disconnected."

    I think we also need that the intersection of their closures is empty, e.g.,

    (-oo,1)\/(1,oo) is not a disconnection of R, since 1 belongs to both their closures,

    IOW, I think we need that the open sets have no limit point in common.
     
  6. Apr 10, 2010 #5

    Office_Shredder

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    The reason it's not a disconnection of R is because neither has 1 in it, not because 1 is in both of their closures.

    But this isn't really relevant, because [tex] (-\infty, 1)[/tex] and [tex](1, \infty)[/tex] is a disjoint cover of the topological space [tex]\mathbb{R}-{1}[/tex]
     
  7. Apr 10, 2010 #6
    O.K, my bad: the actual statement should be that X is disconnected iff (def.)

    X=A\/B , with ClA /\B =empty =A/\ClB , with \/ =union, /\ intersection, Cl=closure

    and A, B subsets of X. And Landau was right.
     
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