Prove that the boundary of an open ball Bra is given by Bra

Prove that the boundary of an open ball Br(a) is given by Br(a) = {x:||x - a||=r}. Prove that Br(a) is a Jord an region for all a R^4 and all r

Solution

(a) May assume a =0 (by translation)

The boundary of a set S is the set of points x, whose every neighbourhood consists of points of S as well as points of S\' (Complement of S)

Let y be a point on the sphere {||x||=r} for r positive.

Then any neighbourhood of x consists of points with ||x|| <r as well as ||x||>r.

This proves the claim.

(b) Br(a) is a Jordan region as it divides Rⁿ into bounded and unbounded disjoint regions .