Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

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.

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
See similar textbooks
icon
Related questions
Question
100%

How do I show 6.13? 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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Paths and Circuits
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS