Let X be an infinite dimensional Banach space and show that the unit sphere

is a

set (i.e. a countable intersection of open sets) in the unit ball

, when the latter is endowed with the relative topology inherited from the weak topology of X.

I know that the unit sphere is dense in the unit ball in the above topology, but how to show it is

? And I know that the open set in X under the weak topology of X are all unbounded, and the complement of unit ball is open since unit ball is closed under the weak topology. Thanks in advance!