= Interior and closure of norm balls
{title2=$\operatorname{int}\overline B(0,1)=B(0,1),\quad\overline{B(0,1)}=\overline B(0,1)$}
In a nonzero <normed vector space>, a unit vector $x$ is approached from outside the unit ball by $(1+\varepsilon)x$, so no boundary point is <interior>. It is approached from inside by $(1-\varepsilon)x$, 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.
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