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Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection.

Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection.

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 6.5? Please explain with great detail.

Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. {Ca}a€a be a collection of subsets of X. Ca. The collection C is an open cover Definition. Let A be a subset of X and let C = Then C is a cover of A if and only if A CU. of A if and only if C is a cover of A and each Ca 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>5, 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 finite subcollection has a non-empty intersection. every This definition can be used in an alternative characterization of compactness.
Transcribed Image Text:Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. {Ca}a€a be a collection of subsets of X. Ca. The collection C is an open cover Definition. Let A be a subset of X and let C = Then C is a cover of A if and only if A CU. of A if and only if C is a cover of A and each Ca 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>5, 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 finite subcollection has a non-empty intersection. every This definition can be used in an alternative characterization of compactness.
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