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Showing the connectedness of a product topology

  1. Oct 6, 2013 #1
    Let A be a proper subset of X, and let B be a proper subset of Y .
    If X and Y are connected, show that X × Y − A × B is connected.

    Attempt: Proven before in my book, I know that since X and Y are each connected that X x Y is also connected. Keeping that fact in mind.

    Pf: Assume (X x Y) - (A x B) has a separation.

    --> (U u V) = (X x Y) - (A x B) where U and V are open disjoint nonempty subsets.

    U and V can each be written in terms of their basis elements of the form (C x D) and (E x F) where C,E [itex]\subset[/itex] X and D,F [itex]\subset[/itex] V

    therefore: (U u V) = (C x D) u (E x F)
    = (C u E) x (D u F)
    --> (C u E) [itex]\subset[/itex] X and (D u F)[itex]\subset[/itex]Y

    Therefore I formed a separation between X and Y, but this cannot occur because we know
    (X x Y) is connected --> Contradiction.


    Now I feel something is a little messed up in the proof, but the idea is that I want to show that
    (X x Y) has a separation which contradicts the fact I know about it.
    Am I even close in my idea?
     
  2. jcsd
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