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Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a point m E C such that for every x E C, x < m.

Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a point m E C such that for every x E C, x < m.

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 6.2? Please explain with great detail.

{Ca}a€a be a collection of subsets of X. Cg. 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 U. 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 ofA. 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. Theorem 6.2. Let C be a compact subset of Rstd. Then C has a maximum point, that is, there is a point m E C such that for every x E C, x < m.
Transcribed Image Text:{Ca}a€a be a collection of subsets of X. Cg. 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 U. 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 ofA. 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. Theorem 6.2. Let C be a compact subset of Rstd. Then C has a maximum point, that is, there is a point m E C such that for every x E C, x < m.
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