Math Forum :: View topic – dense subset

No. If D is a subset of [0,1], define as follows:
for each positive integer n, consider the intervals , where k runs through 0, 1, 2, …, . If such an interval contains a point in D, take a point in the intersection and call that ; if such an interval is disjoint from D, forget it. Then set to be the set of all these (if any). Obviously each is finite; it has at most elements. Also for each point in D, we can find a sequence in converging to it. Hence is a countable dense subset of D.