Given thad X is a compact topological space {Ci}icz, be a collection of non-empty [Ciliez closed Subsets of X that satisfy City CC; +iez. we will prove this by contradiction method. If possible assume that n₁= C₁ = 4 i=] Then and {x\C;};€ open Cover of X is an SiE Z+ By compactness it has a finite Seeb-cover. So UX\C₁ = X for some finite FCZ+ jef But we know ? C; =$, which contradicts lef the C C; for all iezt fact that Cit! Hence our assumption is wrong. So 8 C₁ +4.
Given thad X is a compact topological space {Ci}icz, be a collection of non-empty [Ciliez closed Subsets of X that satisfy City CC; +iez. we will prove this by contradiction method. If possible assume that n₁= C₁ = 4 i=] Then and {x\C;};€ open Cover of X is an SiE Z+ By compactness it has a finite Seeb-cover. So UX\C₁ = X for some finite FCZ+ jef But we know ? C; =$, which contradicts lef the C C; for all iezt fact that Cit! Hence our assumption is wrong. So 8 C₁ +4.
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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![Given thad X is a compact topological space
{Ci}icz, be a collection of non-emply
[Ciliez
closed Subsets of X that satisfy City CC;
+iez.
we will prove this by contradiction method.
&
If possible assume that ₁1 C₁ = 4
i=]
Then
open Cover of X
{x\C;};€
is an
Z+
By compactness it has a finite Seeb-cover.
So U₂₁ X\C; = X for some finite FCZz+
jef
But we know ? C; =$, which contradicts
lef
the C C; for all iezt
fact that Cit!
Hence our assumption is wrong.
So 8. C₁ +4.
$](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F860a1374-20cc-47e2-9366-3835646e97ec%2Fa1c47d48-7f61-497f-81eb-daa50a44d495%2Fikuxsg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given thad X is a compact topological space
{Ci}icz, be a collection of non-emply
[Ciliez
closed Subsets of X that satisfy City CC;
+iez.
we will prove this by contradiction method.
&
If possible assume that ₁1 C₁ = 4
i=]
Then
open Cover of X
{x\C;};€
is an
Z+
By compactness it has a finite Seeb-cover.
So U₂₁ X\C; = X for some finite FCZz+
jef
But we know ? C; =$, which contradicts
lef
the C C; for all iezt
fact that Cit!
Hence our assumption is wrong.
So 8. C₁ +4.
$
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