Metric space question

  • Thread starter jin8
  • Start date
  • #1
jin8
24
0

Homework Statement


the problem:
Let M be a metric in which the closure of every open set is open. Prove that M is discrete


The Attempt at a Solution


To show M is discrete, it's enough to show every singleton set in M is open.
For any x in M, assume it's not open,
then there exist a converging sequence in M-{x} converges to x

I want to show such sequence does not exist, but I really don't know how to use the original statement that the closure of open set is open

Thank for help
 

Answers and Replies

  • #2
Tinyboss
237
0
First show that under this hypothesis, an open set is in fact equal to its own closure. Then with the right choice of open set it's not hard to see that points are open.
 

Suggested for: Metric space question

  • Last Post
Replies
1
Views
880
  • Last Post
Replies
18
Views
3K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
25
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
904
  • Last Post
Replies
9
Views
4K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
6
Views
1K
Top