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Metrizable space

  1. Sep 22, 2010 #1
    hello,
    i have a clarification question:
    if i have a metrizable space, can any open set be written as a countable union of basis elts?
    if not, can it have some properties that make this assertion true, like being T1 or sth like that.
    thanks a lot.

    just to clarify why i think so, i have a space X with a topology T on it.
    Let's say this topology is metrizable, with a metric d. Then, the open sets O(x,r) = {y in X , d(x,y)<r} form a basis for this topology. so any elt in this topology, t in T can be written as a countable union of such O's.
    I just want to know if this is works.
    thx.
     
    Last edited: Sep 22, 2010
  2. jcsd
  3. Sep 22, 2010 #2

    Office_Shredder

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    The answer is no. Take the real numbers with the discrete topology: the distance between any two points is 1. Then consider the set of all positive numbers. Any open ball is one of two things: a single point or all the real numbers. So we're stuck with writing the positive numbers as an uncountable union of singletons
     
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