In a metric space , if closure of every open set is open , then is every singleton open? Let $M$  be a metric space such that the closure of every open set is open , then is it true that every subset of the space is open i.e. that the space is discrete ?

Yes. Suppose $M$ is not discrete. Let $x$ be a non-isolated point, and let $x_1,x_2,\dots$ be a sequence of distinct points in $M\setminus\{x\}$ converging to $x.$ Construct pairwise disjoint open sets $U_n$ with $x_n\in U_n.$ The closure of the set $V=U_1\cup U_3\cup U_5\cup\cdots$ contains $x$ but it contains no neighborhood of $x,$ since it is disjoint from the open set $U_2\cup U_4\cup U_6\cup\cdots.$ Thus $V$ is an open set whose closure is not open.
