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 $\displaystyle A$ be a finite open subset of a metric space $\displaystyle M$. Prove that every point in $\displaystyle A$ is an isolated point of $\displaystyle M$

Definitions and theorems that may come in handy:

Definition:A point $\displaystyle x$ in a metric space is calledisolatedif the set $\displaystyle \{ x \}$ consisting of $\displaystyle x$ 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

Thanks