1. The problem statement, all variables and given/known data In general, in R^n, what is the best way to approach the problem - a given set is open? The given set E is such that for all x,y that belong to the given set, d(x,y) < r. 2. Relevant equations 3. The attempt at a solution let x be the center of the sphere and y be any point such that d(x,y) < r. Now, let z be any boundary point such that d(x,z) = r. Also let d(y,z) < epsilon. We can make a neighborhood N with epsilon as radius and y as point such that all points of N are subset of the given set. In general we can construct a neighborhood N of smallest (of all possible neighborhoods with the same center) radius r , such that N is a subset of E. Hence, all points of the given open set are internal points. Hence, the given set is open. Is it an okay proof? Or should I be proving that the complacent of the open set in a given universe is closed. Hence, the set is open?. I am somewhat new to the method of writing proofs, and so want to know that which is a better way to prove?