I am trying to figure out some properties of density of a set, but I keep confusing myself. I know the definition of dense is: A set E of real numbers is said to be dense if every interval (a, b) contains a point of E. Could I rephrase the definition so that every interval (a, b) contains INFINITELY many points of E? I am trying to justify that by saying that if (a, b) is dense, then there must be a point c in E such that a < c < b. But then (a, c) must be dense, since it is contained in (a, b), and so we could find another point d such that a < d < c, and keep going until we find infinitely many points. But that leads me to another question: If you have two sets, E1 and E2 that are dense, what do you know about E1 intersect E2? Would E1 intersect E2 be dense if E1 is contained in E2? Is there a way to prove the conclusion about the intersection? Also, if a set E is dense, then what do you know about the set A, where E is contained in A? Thanks in advance for any insight!