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Every compact, Hausdorff space is normal.

Every compact, Hausdorff space is normal.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
Transcribed Image Text: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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