For this question I figured that we only need a space that all the subsets are both open and closed at the same time. Besides the discrete space does such a space exist?
2026-05-05 17:39:24.1778002764
Find a metric space that every subset has an empty boundary set
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Let $X$ be such a metric space. Now you want for every $A \subseteq X$ to have $$\partial A = \overline A \setminus A^\circ = \emptyset.$$ As you noted this implies $$\overline A = A = A^\circ.$$ So every set is open. This implies that all subsets of $X$ belong to the topology, so we have the discrete topology which is induced by the discrete metric. Hence, $X$ is discrete.