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Metric space question

  • Thread starter jin8
  • Start date
  • #1
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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
236
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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.
 

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