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Open set in a topological space

  1. Dec 6, 2007 #1
    1. The problem statement, all variables and given/known data
    If U is open set in a topological space such that U = A union B, where A and B are disjoint, do both A and B have to both be open?

    I think that they do not, but I cannot think of a counterexample...perhaps (-1,0) union [0,-1).

    OK. That's a counterexample. So, now the question can both A and B not be open?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Dec 6, 2007 #2


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    What about in R2, trying the same trick only with say, two rectangles, one on (0,1/2)x(0,1) and one on (1/2,1)x(0,1), and then give each half of the boundary between them
  4. Dec 6, 2007 #3


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    Or take U to be (0,1), A=rationals in U and B=U-A. Neither is open.
  5. Dec 7, 2007 #4


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    No need to get as complicated as "rationals" and "irrationals". Take A= [0, 1], B= [2, 3] so that both are closed. Then the intersection is the empty set which is open!

    Or, since the empty set is also closed, you might prefer A= [0, 3/2), B= (1, 2]. Neither is open but their intersection is (1, 3/2) which is open.
  6. Dec 7, 2007 #5
    The union of the two sets is open, not the intersection.
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