I was having "trouble" with two problems. Well, i got a solution for the first (a while ago, i hope i can remember it for today), for the second i was not sure how to approach it.
1) Prove that any set in a metric space is an intersection of open sets.
Yes, a basic question, i know. but that's the problem. my professor said i'm thinking way too hard, and there is a very easy, elegant solution to this.
2) Let be a finite open subset of a metric space . Prove that every point in is an isolated point of
Definitions and theorems that may come in handy:
Definition: A point in a metric space is called isolated if the set consisting of alone is open.
Theorem: A subset of a metric space is open if and only if it is expressible as a union of open balls.
(i think we can use this for the first problem, to prove the claim for singleton points).
Theorem: In a metric space any union of open sets is open.
Theorem: In a metric space a finite intersection of open sets is open.
If any other definitions or theorems are required, i can supply them. just ask. i think this should be enough though