Interior and closure of norm balls ($\mathrm{int}\overline{B}(0,1)=B(0,1),\overline{B(0,1)}=\overline{B}(0,1)$)

In a nonzero normed vector space, a unit vector $x$ is approached from outside the unit ball by $(1+\epsilon )x$, so no boundary point is interior. It is approached from inside by $(1-\epsilon )x$, so every boundary point is in the closure of the open ball. The zero space satisfies the same identities trivially. These statements need the radial structure of a normed space; a two-point discrete metric space with interpoint distance one disproves both analogous claims for general metric balls.