Explore BrainMass
Share

Explore BrainMass

    De Morgan's Laws

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com October 10, 2019, 1:58 am ad1c9bdddf
    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.

    $2.19