Assignment:
Q1. Let X be a set and T and T' are two topologies on X.
Prove that if T subset of T' and (X,T') is compact, then (X,T) is compact.
Prove that if (X,T) is Hausdorff and (X,T') is compact with T subset of T', then T=T'.
Q2. Let X be a topological space. A family {F_a} with a in I of subsets of X is said to have the finite intersection property if for each finite subset J of I, the intersection of F_a with a in J is not empty .
Prove that X is compact if and only if for each family {F_a} with a in I of closed subsets of X that has the finite intersection property, the intersection {F_a} with a in I is not empty.
Provide complete and step by step solution for the question and show calculations and use formulas.