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Determining whether a set is open, closed, or neither

  1. Sep 17, 2015 #1
    1. The problem statement, all variables and given/known data
    Determine whether the given set is open or closed (or neither):
    {(x,y,z) ε R^3, 1<x^2+y^2<4}

    2. Relevant equations


    3. The attempt at a solution
    I believe that the set is open due to the only bounds being inequalities that are less than, not less than or equal to, but that seems like an oversimplification of the problem. Am I missing something? Thanks
     
  2. jcsd
  3. Sep 17, 2015 #2

    andrewkirk

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    To prove the set is open you need to prove that, given a starting group of subsets of the space (a 'basis') the set can be constructed from a collection of sets in the basis via only the operations of unions and a finite number of intersections.

    In ##\mathbb{R}^2## the usual basis is either:

    1. the collection of all open rectangles ##\{(a,b)\times (c,d)\ | \ a,b,c,d\in\mathbb{R}\}##; or
    2. the collection of all open balls ##\{B_\epsilon(\mathbf{x})\ | \ \mathbf{x}\in\mathbb{R}^2\wedge \epsilon>0\}## where ##B_\epsilon(\mathbf{x})## is the set of all points with distance less than ##\epsilon## from ##\mathbf{x}##

    Can you write your set as a union of sets exclusively from one or the other of those collections?
     
    Last edited: Sep 17, 2015
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