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

(a) Prove that the boundary of an open ball Br(a) is given by Br(a) ={x : ||x - a|| = r}.

(b) Prove that Br(a)is a Jordan region for all a in Rn and all r> 0.

Solution

(a) (b) The boundary of a set S is the set of points y such that every neighbourhood of y contains points of S as well as S\' (S\' being the complement of S)

We may take a =0 without loss of generality (by translation to the origin).

The set of points {y: ||y||=r} clearly meets the definition of the boundary set out above.

It also separates the Euclidean space into two connected components , one bounded and the other unbounded.

So Br(0) (hence Br(a)) is a Jordan region