Problem Statement: (a) Given a set define what it means for a point to be an interior point of . (b) For let . Show that is open in .
(a)Definition: A point is an interior point of if there exists a real number such that .
(b) Proof: We wish to show that every point of is an interior point of . Let , then is an interior point of and so there exists some such that .
Now pick and define to be the largest possible value such that . We know such a exists since is an open set. Then it follows that , thus, . Since was an arbitrary point in it follows that and so is an isolated point of . Therefore, is open.
Reflection: Straight from the definitions.