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Intersection of open sets

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data
    a) If U and V are open sets, then the intersection of U and V (written U \cap V) is an open set.
    b) Is this true for an infinite collection of open sets?


    2. Relevant equations

    Just knowledge about open sets.


    3. The attempt at a solution

    a) Let U and V be open sets, x be in U \cap V, and a,b be real numbers. Then there exists open intervals (a1,b1) in U and (a2,b2) in V such that x is in (a1,b1) and x is in (a2,b2). Now let (a,b) = (a1,b1) \cap (a2,b2). Since x is in (a1, b1) and x is in (a2,b2), then x is in (a,b) and by definition is an open interval. Moreover, since (a,b) is a subset of U \cap V, then U \cap V is an open set.

    Is there anything I'm missing from this proof, like proving that the intersection (a1,b1) \cap (a2,b2) is an open interval? Or is that overdoing it..?

    b) I know this isn't true (the intersection from n=1 to infinity of all open sets (-1/n, 1/n) = {0}, which is not an open set), but again I feel like I'm missing something - specifically proving that {0} is the intersection of all of those open sets, and proving {0} is not open.

    I know these questions are incredibly straightforward, but my professor is kind of a stickler on the little details, especially the ones that seem pretty obvious. Thanks for the help!
     
  2. jcsd
  3. Nov 29, 2011 #2

    micromass

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    This is good. However, it would be nice to state in the beginning that you are working with open sets in [itex]\mathbb{R}[/itex]. Your argument does not hold for other spaces (as the open sets there are not necessarily generated by intervals).

    This is crucial. I would include it.

    OK, you got the point. Now, where are you stuck? On proving that intersection, or proving that {0} isn't open??

    By the way, it is possible to type LaTeX here. See this thread https://www.physicsforums.com/showthread.php?t=546968
     
  4. Nov 29, 2011 #3
    I'm basically stuck on proving the intersection (for both parts a and b) and proving 0 isn't open. I think I need just a shove in the right direction..they seem so self-explanatory to me that I don't know how to go about proving any of these things too rigorously.
     
  5. Nov 29, 2011 #4

    micromass

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    OK, let's start by proving that {0} isn't open. We need definitions for this. How did you define open??
     
  6. Nov 29, 2011 #5
    A set S is open if for all x in S there is an open interval (a,b) contained in S with x in (a,b).

    Could we say that for any ε > 0 there is no open interval (0-ε, 0+ε) that is contained within {0}, so {0} isn't open? Or something like that?
     
  7. Nov 29, 2011 #6

    micromass

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    Good. Let's apply this definition on {0}. Clearly, our x=0. So we must find an open interval [itex](a,b)\subseteq \{0\}[/itex] such that 0 is in (a,b). But then [itex](a,b)=\{0\}[/itex] has one element. Can you derive a contradiction from this?? (for example, by showing that (a,b) has more than one element).

    For the intersection questions. You need to prove

    [tex](a,b)\cap (c,d)=(\max\{a,c\},\min\{b,d\})[/tex]

    and

    [tex]\bigcap_{n\in \mathbb{N}_0}{(-1/n,1/n)}=\{0\}[/tex]

    These are just equalities of sets. Do you know how to show an equality of sets?? To prove A=B, just pick an x in A and show that it is in B and pick an x in B and show that it is in A.
     
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