Weak-star open dual ball category obstruction ($\{\ell \in E^{\ast }:\Vert \ell \Vert <1\}={\bigcup }_n\{\Vert \ell \Vert \le 1-1/(n+1)\}$)

For an infinite-dimensional normed space, the weak-star open unit ball of its dual is not a Baire space. The smaller closed norm balls are weak-star closed and have empty relative interior: a nonzero functional annihilating any finitely many tested vectors can move the norm beyond the smaller radius while remaining below one. Their countable union is the open norm ball. Thus this topology is not topologically complete, regardless of separability.