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Trivial set theory question?

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data

    Suppose X ⊂ R^n is a compact set, and U_1, U_2, U3, ... ⊂ R^n are open sets whose union contains X. Prove that for some n ∈ N (the natural numbers) we have X ⊂ U_1 ∪ ... ∪ U_n.

    2. Relevant equations

    A set is called compact if it is both closed and bounded.

    3. The attempt at a solution

    This problem seems trivial to me. If, as stated in the problem, U_1, U_2, U3, ... ⊂ R^n are open sets whose union contains X, does that mean that for some n we have X ⊂ U_1 ∪ ... ∪ U_n? I don't understand how there is anything to prove here. Any help would be appreciated.
  2. jcsd
  3. Dec 2, 2008 #2
    You have an infinite union of open sets which cover some compact set X, you wish to show that finitely many of them suffice to cover X.
  4. Dec 2, 2008 #3
    Ah, ok. That makes a lot more sense. Would this proof suffice?

    We will prove by contradiction. Assume that no finite number of the U's will contain all of X. Let x_k ∈ X be one such element that is cannot be contained in a finite number of the U's. But if x_k ∈ X, then it must be contained in at least one of U_1, U_2, U_3 .... So assume that x ∈ U_k. But if we adjoin U_k to our list of U's that earlier did not contain x_k, then it will still be finite and will now contain x_k. We can repeat this procedure for all x_k to show that the list of U's necessary to contain x_k must be finite.

    Upon further review, I don't think that this can be right. If we have an infinite number of x_k's, then our set of U's would also be infinite...
  5. Dec 2, 2008 #4
    This makes intuitive sense to me, but I'm not sure if it would work as a proof:

    Since each U is open, there is an open ball around each point in U. But an open ball must have some finite volume, and since the set X is compact, it can be contained by a ball around the origin, and it must thus have finite volume also. Since X has finite volume and each U has finite volume, there must be a finite number of U's that can completely contain X.

    Can I use the term "Volume" to describe the space contained by a ball in R^n? Also, how can I make this intuitive proof rigorous?
  6. Dec 2, 2008 #5
    You don't know this. The cylinder [tex]\{ (x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 < 1 \}[/tex] is an example of an open set that doesn't have finite volume. There are just infinitely many of those "finite-volume" open balls.

    You'll need to use the fact that X is closed (and bounded), because the condition given in the problem is equivalent to compactness in R^n.
  7. Dec 3, 2008 #6


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    Staff Emeritus
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    By the way, not only is this not a "trivial" problem, it is not a problem in set theory: set theory does not include the concepts of "open", "closed", or "compact" sets. This is topology.
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