Proving the Discreteness of a Metric Space with Open Closure Property"

[jin8]

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

 

[Tinyboss]

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.