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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_04” 2011/5/22 23:17 page 31 #31
Topological Aspects of the Real Line 4-31
we can get a finite sub-cover for S using the finite number of these
balls of radius
δ
3
(which covers S) and the corresponding G
α
s. Hence
S is compact.
In the following we shall obtain certain properties of compact
subsets in a metric space, which will be useful later.
Theorem 4.3.31 Let (X , d) be a metric space. If A X is compact
and B X is closed with A B =∅, then d(A, B)>0.
Proof Since d(A, B ) 0 always, we shall assume d(A, B) = 0 and
deduce a contradiction to the hypothesis. Indeed if d(A, B) = 0 then
from the definition and Theorem 4.2.18,
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Publisher Resources

ISBN: 9781299447561Publisher Website