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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 36 #36
4-36 Real Analysis
Theorem 4.3.43 Every non-empty open subset of R
k
is at most a
countable disjoint union of open connected sets.
Proof By Theorem 4.3.40 (iv), any non-empty open subset U of
R
k
(considered as a metric space) can always be written as a disjoint union
of maximal connected sets namely its components. First we shall show
that these components are open. Indeed, if C is a component and x C,
we can always take a neighbourhood of x (a ball with center x radius
δ>0), which is completely contained in U . In as much as this ball
is connected (indeed, if B(x, δ) is disconnected, then we can
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Publisher Resources

ISBN: 9781299447561Publisher Website