Closed sets in metric spaces

Question

I have just proven any closed set in the plane (R^2) with the usual metric is the boundary of some subset of the plane.

I am now struggling to find a counterexample to show that this is not true for any metric space. Thanks for your time.

Answer 1

Hint: Use the standard source of counterexamples: a discrete metric space.

Answer 2

Consider the space $E = \lbrace 0, 1 \rbrace$, endowed with the induced metric.

The subset $\lbrace 0 \rbrace$ is closed, but is not the boundary of any subset.