-Re call, Ac (Xicd) AB .bounded if JxcX,ryo Sucb that Ac BCX) A B(メイ) property (Xdl) methic space AcX, A Compact insX, JaJ → bounded A is prof. in (X,d) Hint: tette x crbitrary in X Idnd BX,n).n=1, 2,3. open Cover of X A Compuct Conclude # Show

Elementary Linear Algebra (MindTap Course List)
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Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 37E: Let V be the set of all positive real numbers. Determine whether V is a vector space with the...
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Becall AccC X,ch) AB :bounded if
JACX,r yo Sucb tha
Ac BCXX)
property (Xdl) metric epace
Ac X, A Compact in CX, Ja J A is
bounded
proof.
in (X,d)
Hint tetie x carbitrary
Idnd BX,n).nEl, 2,3.
Show opan Cover of X
A Compuct
in
uSG
condlude
Transcribed Image Text:Becall AccC X,ch) AB :bounded if JACX,r yo Sucb tha Ac BCXX) property (Xdl) metric epace Ac X, A Compact in CX, Ja J A is bounded proof. in (X,d) Hint tetie x carbitrary Idnd BX,n).nEl, 2,3. Show opan Cover of X A Compuct in uSG condlude
Definition: A space X is compact provided that every open cover of X has a finite
subcover. Equivalently, X is compact provided that for every collection O of open
sets whose union equals X, there is a finite subcollection {O;}\1 of O whose union
equals X. A subspace A of a space X is compact provided that A is a compact
topological space in its subspace topology.
Transcribed Image Text:Definition: A space X is compact provided that every open cover of X has a finite subcover. Equivalently, X is compact provided that for every collection O of open sets whose union equals X, there is a finite subcollection {O;}\1 of O whose union equals X. A subspace A of a space X is compact provided that A is a compact topological space in its subspace topology.
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