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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 40 #40
4-40 Real Analysis
A collection τ of subsets of X can now be defined as follows.
B τ if and only if either B is empty or for each x B there exists
U
C such that x U B. Then (X , τ) is a topological space. The
collection
C will also be called a basis for τ .
Proof The required axioms can be easily verified.
Note 4.3.54 Consider the extended real number system [−∞, ∞] =
R {−∞, +∞}. Let C be the collection of all intervals of the form
(a, b) with −∞ a < b ≤∞together with the sets of the form
[−∞, a) = (−∞, a) {−∞} (a
R) and (b, ∞] = (b, ) {+∞}
(b
R). Obviously, C is a collection
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Publisher Resources

ISBN: 9781299447561Publisher Website