- #1

Adorno

- 30

- 0

## Homework Statement

Let X be a metric space and A a subset of X. Prove that the following are equivalent:

i. A is dense in X

ii. The only closed set containing A is X

iii. The only open set disjoint from A is the empty set

## Homework Equations

N/A

## The Attempt at a Solution

I can prove that i implies ii: assume that there is a closed set B containing A other than X, and show that B must equal X since the closure of A is X.

Presumably I can prove the other implications (ii -> iii and iii -> i) in a similar way but I'm not sure how to get started. Is there a certain property or fact I should be using?