How do I show 6.12? Please explain with great detail. Theorem 6.12. Every compact, Hausdorff space is normal.Theorem 6.13. Let B be a basis for a space X. Then X is compact if and only if everycover of X by basic open sets in B has a finite subcover.Definition. Let A be a subset of X and let C = {C«}gea be a collection of subsets of X.Then C is a cover of A if and only if A cUrca Ca. The collection C is an open coverof A if and only if C is a cover of A and each C, is open. A subcover C' of a cover C ofA is a subcollection of C whose elements form a cover of A.For instance, the open sets {(-n, n)}nen form an open cover of R. A subcover ofthis cover is {(-n, n)}n>s, because these sets still cover all of R.Definition. A space X is compact if and only if every open cover of X has a finitesubcover.Definition. A collection of sets has the finite intersection property if and only ifevery finite subcollection has a non-empty intersection.Theorem 6.8. Let A be a closed subspace of a compact space. Then A is compact.Theorem 6.9. Let A be a compact subspace of a Hausdorff space X. Then A is closed.

To prove: Every compact, Hausdorff space is normal. Let X be a compact Hausdorff space. Let A,B⊂X be…

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Every compact, Hausdorff space is normal.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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How do I show 6.12? Please explain with great detail.

Theorem 6.12. Every compact, Hausdorff space is normal.
Theorem 6.13. Let B be a basis for a space X. Then X is compact if and only if every
cover of X by basic open sets in B has a finite subcover.
Definition. Let A be a subset of X and let C = {C«}gea be a collection of subsets of X.
Then C is a cover of A if and only if A cUrca Ca. The collection C is an open cover
of A if and only if C is a cover of A and each C, is open. A subcover C' of a cover C of
A is a subcollection of C whose elements form a cover of A.
For instance, the open sets {(-n, n)}nen form an open cover of R. A subcover of
this cover is {(-n, n)}n>s, because these sets still cover all of R.
Definition. A space X is compact if and only if every open cover of X has a finite
subcover.
Definition. A collection of sets has the finite intersection property if and only if
every finite subcollection has a non-empty intersection.
Theorem 6.8. Let A be a closed subspace of a compact space. Then A is compact.
Theorem 6.9. Let A be a compact subspace of a Hausdorff space X. Then A is closed.

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