# De Morgan's Laws

1.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'.

2.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.

https://brainmass.com/math/geometry-and-topology/de-morgans-laws-fip-hausdorff-compact-357449

#### Solution Preview

1. Let X be a set and T and T' be two topologies on X. Prove that if T is a subset of T' and ( is compact, then is compact.

Prove that if is Hausdorff and is compact with T subset of T', then

Proof: Let X be a set and T and T' be two topologies on X. Suppose and is compact.

Let be an open cover of X by open sets in T. So Since every is an open set in So is an open cover of X by open sets in Since is compact, there exists a finite subcollection such that Therefore, is compact.

Now suppose that is Hausdorff and is compact with Consider the identity function defined by for all ...

#### Solution Summary

De Morgan's Laws are applied. FIP, Hausdorff and Compact are examined.