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Proving open and closed sets

  1. Sep 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Let {Ei: 1≤i≤n} be a finite family of closed sets. Then ∪i=1n Ei is closed.

    2. Relevant equations
    Noting that (Ei)c is open

    3. The attempt at a solution
    Honestly, I have no idea where to start.

    I tried to demonstrate that Eai≥Ei if a is a constant greater than zero. Then showing that Eai is closed which means that Ei is closed.
     
  2. jcsd
  3. Sep 29, 2015 #2

    FeDeX_LaTeX

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    This is easiest to do if you instead try showing that, if [itex]E_i[/itex] are open sets, where [itex] 1 \leqslant i \leqslant n[/itex], then [itex]\bigcap_{i=1}^{n} E_{i}[/itex] is open, which pretty much follows from the definition of an open set.

    You can then take complements and you're done.
     
  4. Sep 30, 2015 #3

    Fredrik

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    When you have more experience with proofs, you will find it very easy to see where to start in problems like this. There's a statement P that you're supposed to use to prove a statement Q. In this situation, you should almost always ask yourself what the definitions of the terms in statement Q are telling you that statement Q means.

    What does it mean to say that ##\bigcup_{i=1}^n E_i## is closed? The answer depends on your definition of closed. The most common definition is "a set is said to be closed if its complement is open". If that's your definition, the answer is: ##\left(\bigcup_{i=1}^n E_i\right)^c## is open. The proof of that is rather trivial, if you know the most basic theorems.

    If you're working with metric spaces, perhaps your definition of closed is something else, e.g. "a set E is said to be closed if the limit of every convergent sequence in E is in E". Then the definition suggests another strategy: Start by saying "Let ##(x_n)_{n=1}^n## be a convergent sequence in ##\bigcup_{i=1}^n E_i##". Then try to prove that ##\lim_n x_n\in \bigcup_{i=1}^n E_i##.
     
  5. Sep 30, 2015 #4

    HallsofIvy

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    As both FedExLatex and Fredrick said, how you prove something often depends on how it is defined! Exactly how are open and closed sets defined in your course?
     
  6. Sep 30, 2015 #5
    Here are the definitions my professor provided us
    definition.png
     
  7. Sep 30, 2015 #6

    HallsofIvy

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    Okay,, a set is closed if and only if its complement if open. That was the first definition Fredrick gave. I notice you did NOT define "open set". In general "point set" topology, a "topology" for a set, X is a collection of subsets of X such that, X itself is in the collection, the empty set is in the collection, the union of any number of sets in the collection is also in the collection and the intersection of any finite number of sets in the collection is also in the set. To show that "the union of a finite number of closed sets is closed", using that definition, you have to show that its complement is open. To do that you can use the fact that [itex]\left(\cup_{n} A_n\right)^c= \cap_{n}A_n^c[/itex].
     
  8. Sep 30, 2015 #7
    Thank you, this helps a great deal

    cheers
     
  9. Sep 30, 2015 #8

    Fredrik

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    Since the definitions you quoted specifically mentioned ##\mathbb R^n##, rather than some arbitrary topological space, it's very likely that your book's definition of open is "a set E is said to be open if every element of E is an interior point of E". This should tell you how to proceed with the proof.
     
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