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Open and closed sets of metric space

  1. May 12, 2010 #1
    1. The problem statement, all variables and given/known data
    I am using Rosenlicht's Intro to Analysis to self-study.

    1.) I learn that the complements of an open ball is a closed ball. And...
    2.) Some subsets of metric space are neither open nor closed.

    2. Relevant equations

    Is something amiss here? I do not understand how both can be true at the same time.
     
  2. jcsd
  3. May 12, 2010 #2

    HallsofIvy

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    No. The complement of an open set is an closed set but the complement of a "ball" is not a "ball". An open ball is of the form [itex]B_r(p)= \{ q| d(p, q)< r\right}[/itex]. In R, an "open ball" is an open interval, (a, b). Its complement is [itex](-\infty, a]\cup [b, \infty)[/itex] which is closed but not a "ball".

    I don't see what one has to do with the other. The complement of any open set is closed, the complement of any closed set is open. The complement of a set that is neither closed nor open is neither closed nor open. The "half open interval" in R, (0, 1], is neither closed nor open.

    By the way, there also exists sets in a metric space that are both open and closed!
     
  4. May 12, 2010 #3
    But I know that in any metric space, an open ball is an open set/ closed ball is a close set. Also, the complement of an open set is a closed set.

    But then according to you,

    The complement of an open ball is not closed ball.

    So an open set is not an open ball?
     
  5. May 12, 2010 #4

    jbunniii

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    There are open sets that are not open balls. For example, a set consisting of the union of two disjoint open balls is an open set, but it is not an open ball.

    An open ball is a set consisting of all points less than a certain distance from a given point.

    An open set is any set with the following property: no point is so close to the "boundary" that I can't center a suitably small open ball around that point, such that the ball is entirely contained in the set.

    Every open ball is an open set but not vice versa.
     
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