Another countable dense subset problem

  • Thread starter radou
  • Start date
  • #1
radou
Homework Helper
3,115
6

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.
 

Answers and Replies

  • #2
22,089
3,297
Completely correct!!
 
  • #3
radou
Homework Helper
3,115
6
Excellent, thanks!
 

Related Threads on Another countable dense subset problem

Replies
9
Views
2K
Replies
6
Views
8K
Replies
4
Views
929
  • Last Post
Replies
7
Views
6K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
5
Views
7K
Replies
3
Views
2K
Replies
11
Views
3K
Replies
3
Views
1K
Top