Prove: Interior is an open set
Let (X,d) be a metric space and let F be a subset of X. B(r,x) is the open ball of radius r about x.
Definition: The interior of F, int(F), is the set of all x E F such that there is an r > 0 such that B(r,x) is contained in F.
Definition: Let D be a subset of X. By definition, D is open iff for all a E D, there exists r>0 such that B(r,a) is contained in D.
Using these definitions, prove that int(F) is an open set.
Let x E int(F), then there exists r>0 such that B(r,x) is contained in F.
We need to prove that there exists some r' >0 s.t. B(r',x) is contained in int(F). How can we prove this?
Any help is greatly appreciated!