1. The problem statement, all variables and given/known data 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 2. Relevant equations N/A 3. 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?