1. Prove that Cl(Q) = R in the standard topology on R.

2. Let B = {a/2^n R | a Z, n Z+}. Show that B is dense in R.

3. Determine the set of limit points of A = {1/m + 1/n R | m,n Z+} in the standard topology on R.

4. Let T be the collection of subsets of R consisting of the empty set and every set whose those complement is countable.

a) Show T is a topology on R. (It is called the countable complement topology)

b) Show that the point 0 is a limit point of the set A = R - {0} in the countable complement topology.

c) Show that in A = R - {0} there is no sequence converging to 0 in the countable complement topology.