boundary points of an infinite subset of a metric space

Question

Does any infinite subset of a metric space have boundary points ?

I know that the set of boundary points of a metric space is empty.But i am not very sure about whether, this is true for any infinite subset of a metric space.

Suppose S is an infinite subset of N, then will it have any boundary points ?

I am confused, i would really appreciate some help.

Answer 1

In a discrete space, every subset has empty boundary, since every subset is both open and closed.

Generally, a subset in a topological space has empty boundary if and only if it is both open and closed, since we have $\partial A = \overline{A}\setminus \overset{\circ}{A}$.