Another countable dense subset problem

  1. radou

    radou 3,215
    Homework Helper

    1. The problem statement, all variables and given/known data

    This one seems pretty simple - that's exactly why I want to check it.

    One needs to show that if X has a countable dense subset, then every collection of disjoint open sets in X is countable.

    3. The attempt at a solution

    Let U be a collection of disjoint open sets in X, and let A be a countable subset of X which is dense in X.

    Let Ui be any member of U. Then Ui contains an element of A, since if x is in Ui, x is either in A or a limit point of A. If x is a limit point of A, choose a basis element of the topology of X which contains x and is contained in Ui. Then this basis element intersects A in some point other than x. Hence, Ui contains an element of A.

    Since any two members of U are disjoint, they will contain different elements from A. This gives a one-to-one correspondence between the countable set A and the family U, so U is countable.
     
  2. jcsd
  3. micromass

    micromass 19,188
    Staff Emeritus
    Science Advisor
    Education Advisor

    Completely correct!!
     
  4. radou

    radou 3,215
    Homework Helper

    Excellent, thanks!
     
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