Download The Cantor Ternary Set: The word ternary means third and indicates

The Cantor Ternary Set:
The word ternary means third and indicates that at each stage the
middle third of each remaining open interval is deleted .The name Cantor
honors the Russian-born German mathematician Goreg Cantor, he
developed the theory of sets a century ago. His work had a profound
effect on the logical basis for theorems of Calculus and higher
mathematics .
Let C be a subset of the interval [0,1]
We define C by first selecting the two closed subintervals
of length ,each of them is closed set then
second selecting the four closed subintervals
then
is closed set,
,
of length
is closed set.
The process continues indefinitely .At the n-th stage we select
subintervals
Then the set
,
L
n
,…,

closed
of length
…
is closed.
Then we define (as follows)
C= the collection of all points in [0,1] that are in each
n=1,2,3,…that is
The set C is called the Cantor ternary set
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L for
n
Cantor set have many topological properties :
Definition
A set D is said to be closed if it contains all its limit points .In other
words, a set D is said closed if whenever
is a sequence of points
in D such that lim i xi  x then x is also in D.
Examples
.D=[0,1] is closed
. Any finite point set is closed
.The union of a finite number of closed intervals
.S=
is closed
. Q  [0,1] not closed because the sequence 0.5, 0.505 , 0.5055 , 0.505505
,…,the limit point of this sequence is irrational number .
Lemma (1)
C is closed
Proof
n  N ,the set
L is closed ,since it is the union of a finite collection of
n
closed intervals.
Let
be a sequence in C with lim i xi  x .
By definition of C ,the sequence in the intersection
So the sequence in
L , n  N
n
2
Since
L closed then
n
the it contains all its limit points, i.e
x  Ln , n  N
Thus x C
Therefore , every limit point of a sequence in C is in C ,so that C is closed
Definition
A set D is said to be totally disconnected if it contains no nonempty open
intervals .
Examples
. Any finite point set
.the rational set in the interval (0,1)

. the set 1 n : n  N

Lemma (2)
C is totally disconnected
Proof:
Since
L consists of intervals of length
n
and the length
and since C  Ln for each n ,
converge to zero when n converge to infinity, we
conclude from the remark preceding the lemma that C cannot contain any
non-empty open intervals . Thus C is totally disconnected. 
Definition
A set D of real numbers is said to be perfect provided that it is closed
and each its points is a limit of other points in D.
Examples
.Any closed interval with more one point is perfect
.Any closed set with isolated points is not perfect
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Lemma (3)
C is perfect
Proof:H.W
Theorem
C is closed ,bounded , totally disconnected and perfect subset of [0,1]
.
Proof:
From lemmas (1),(2)and (3). 
Definition
A set D of real numbers that is closed , totally disconnected and perfect is
called a Cantor set.
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