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Nov19-10, 12:51 PM
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.
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