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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 1 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 3 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. 4