I have a problem in my book, which asks me
Prove that, in a discrete metric space $(X,d)$, $\partial A=\emptyset\quad\forall A\subseteq X$. Prove that, $(X,d)$ is a discrete metric space.
Then, the author asks - Is the converse true? i.e. If in a metric space $(X,d)$, $\partial A=\emptyset\quad\forall A\subseteq X$, then is $(X,d)$ a discrete metric space?
Can anybody help me in this regard? Thanks for your assistance in advance.
N.B. Here $\partial A$ denotes the set of all boundary points of $A$ i.e. set of all those points in $X$ which are not interior nor exterior point of $A$.
If $X$ is not discrete, then there's non-closed subset $A$ of $X$. Take $x\in\overline A\setminus A$. Then $x\in\partial A$ and therefore $\partial A\neq\emptyset$.