doubt in a book proof from 'The Geometry of Fractal Sets'

Question

I am reading the proof of existence of positive finite $H^s$-measure (Theorem 5.4) on page 67-68 of The Geometry of Fractal Sets.I am not quite convinced that $E_k$ are closed set by the construction in the proof and I think it is not trivial to prove the $E_k$'s are closed in the case when $n>2$. Can someone explain it to me?Thnak you.