Another countable dense subset problem

[radou]

Homework Statement

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.

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.

 

[micromass]

Completely correct!

 

[radou]

Excellent, thanks!