1. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?

2. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

3. Let X be a set and F is separating collection of functions f: X --> Y_f, each from X into a topological space Y_f. Prove that X with the weak topology by F is Hausdorff.

4. Explain how we can think of the unit sphere (a subspace of IR^3 with its usual topology) as a subset of the Hilbert cube.