Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Metrizable space

  1. Sep 22, 2010 #1
    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.
    Last edited: Sep 22, 2010
  2. jcsd
  3. Sep 22, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook