Page 71 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

a sufficient condition for a set to possess a limit point. It states that an infinite and bounded Notes
set must have a limit point. This condition is not necessary in the sense that an unbounded
set may have a limit point.
 The limit points of a set may or may not belong to the set. However, if a set is such that
every limit point of the set belongs to it, then the set is said to be a closed set. The concept
of a closed set has been discussed. Here, we have also shown a relationship between a
closed set and an open set in the sense that a set is closed if and only if its complement is
open. Further, we have also defined the Derived set of a set S as the set which consists of all
the limit points of the set S. The Union of a given set and its Derived set is called the closure
of the set. Note the distinction between a closed set and the closure of a set S.
 Finally, we have introduced another topological notion. It is about the open cover of a
given set. Given a set S, a collection of open sets such that their Union contains the set S is
said to an open cover of S. A set S is said to be compact if every open cover of S admits of
a finite subcover. The criteria to determine whether a given set is compact or not, is given
by a theorem named Heine-Borel Theorem which states that every closed and bounded
subset of R is compact. An immediate consequence of this theorem is that every bounded
and closed interval is compact.

3.9 Keywords

Bulzano Weierstrass Theorem: Every infinite bounded subset of set R has a limit point (in K).
Compact Set: A set is said to be compact if every open cover of it admits of a finite subcover of
the set.

Heine-Borel Theorem: Every closed and bounded subset of R is compact.
3.10 Review Questions

1. Prove that –|x| = Min. {x, – x} for any x R. Deduce that –|x|  x, for every |x|R.
Illustrate it with an example.

2
2
2. Prove that |x| = x , for my xR.
3. For any two real numbers x and y (y ), prove that
x x
= .
y y
4. Prove that |x – y|  ||x| – |y|| for any real numbers x and y.

5. Test which of the following are open sets:
(i) An interval [a, b] far aR, b R, a < b
(ii) The intervals [0, l [; and ] 0, 1[
(iii) The set Q of rational numbers
(iv) The set N of natural numbers and the set Z of integers.

{ 1 }
(v) The set n : n  N

(vi) The intervals ]a, [ and [a, [ for aR.

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