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Homework Help: Connectedness in topology

  1. Feb 24, 2010 #1
    Let A, B be two connected subsets of a topological space X such that A intersects the closure of B .
    Prove that A ∪ B is connected.

    I can prove that the union of A and the closure of B is connected, but I don't know what to do next. Could anyone give me some hints or is there another way to treat this problem?

    Also, for the specific case, the union of the open ball B((−1, 0), 1) and closed ball B((1, 0), 1) should be connected, right? I can see it, but I'm not sure how to word the proof. Any help is greatly appreciated!
     
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  3. Feb 24, 2010 #2

    Dick

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    I would assume AuB is disconnected and try to find a contradiction with A intersecting the closure of B.
     
  4. Feb 24, 2010 #3
    Let [tex]A \cup B = E_1 \cup E_2[/tex], with [tex]\overline{E_1} \cap E_2 = \emptyset[/tex] and [tex]\overline{E_2} \cap E_1 = \emptyset[/tex]
    (the overline denoting closure). You need to prove one of E_1, E_2 is the empty set.
     
  5. Feb 24, 2010 #4
    A space is disconnected means that it's the union of two disjoint nonempty open sets. Since we're given that A,B are connected, the only way we could possibly write A u B as a union of disjoint open sets is if A and B are in fact open and disjoint.

    You can see this can't be true since A intersects the closure of B, i.e. there's a point a in A such that any neighborhood of a intersects B. But if A is open, then A contains a neighborhood of a, so that A intersects B.
     
  6. Feb 25, 2010 #5
    Thanks a lot for your guys. I wasn't that clear about the concept in the first place, but now I know how to handle this type of problems.
     
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