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 thefinite intersection property has a non-empty intersection.{Ca}a€a be a collection of subsets of X.Ca. The collection C is an open coverDefinition. 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 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>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 finitesubcover.Definition. A collection of sets has the finite intersection property if and only iffinite subcollection has a non-empty intersection.everyThis definition can be used in an alternative characterization of compactness.

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Answered: Theorem 6.5. A space X is compact if…

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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 26E: Let A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to...

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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.

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