(4 marks) Let X be a non-empty set equipped with the discrete topology. Prove that X is compact if and only if it is finite.
(4 marks) Provide an example of two connected sets A and B such that their intersection A Ո B is not connected. A sketch and an explanation of your answer is sufficient.
(4 marks) Assume that f: [a,b] → [a, b] is a homeomorphism. Prove that f must send the point a to one of the end points, i.e., that f satisfies f(a) = a or f(a) = b.