Is (X, U) compact? (X, T)?

Question

Let T and U be topologies on a set X.

a.) Suppose (X,T) is compact and T is contained in U. Is (X,U) compact?

b.) Suppose (X,U) is compact and T is contained in U. Is (X, T) compact?

c.) Suppose (X,T) and (X,U) are compact Hausdorff spaces. Show either T=U, T is not contained in U, or U is not contained in T.

This is a problem in my textbook that I came across while studying for my topology exam. I really have no clue and was hoping for some helpful hints, clues, or explanations. Thanks!

Answer 1

a and b are false, as can be shown with simple counterexamples. Look at $\mathbb{R}$ with various topologies.