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