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Open set (equivalent definitions?)

  1. Jan 2, 2014 #1
    I've seen open sets ##S## of a bigger set ##X## being defined as

    1) for every ##x\in S## one can find an open disk ##D(x,\epsilon)## centered at ##x## of radius ##\epsilon## such that ##D## is entirely contained in ##S##. Where

    $$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$
    and ##d## is a metric.

    2) An open set is a set that can be written as a union of open disks.

    Are these two definitions equivalent in general? Or does it require ##X## to be Hausdorff. If they are in general equivalent, can you outline a proof?
     
  2. jcsd
  3. Jan 2, 2014 #2
    I found a proof here:
    http://people.hofstra.edu/stefan_waner/diff_geom/openballs.html [Broken]
     
    Last edited by a moderator: May 6, 2017
  4. Jan 2, 2014 #3
    The two definitions are equivalent if the topological space in question is metrizable.

    I recommend proving for yourself that every metric space satisfies the Hausdorff axiom.
     
  5. Jan 4, 2014 #4

    WWGD

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    More generally, the open subsets of a topological space ##X ## are , or can be, any collection of subsets of ## X## that are closed under unions and closed under finite intersection, and the collection includes the whole space ## X## and the empty set. You then have a sub -collection of the collection of open sets that is called a basis, so that for every element ##x ## is an open set ## U## , there is a basis element ##B## with ## x ## contained in ## B ##, and ## B \subset U ##.
     
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