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 coverDefinition. 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 ofA 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 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.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.

We start the proof by assuming there are no such maximum element in C.

Answered: Theorem 6.2. Let C be a compact subset…

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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 4E: Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an...

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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.

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