CANTOR SET

                                                             CANTOR SET

                INTRODUCTION :

                                     We have learn about open sets and its characterization on the real line.  We now consider the structure of its closed sets. Among the simplest closed sets on the real line are the closed intervals and finite union of closed intervals. Finite sets are also include the single point set which is a closed interval with equal end-points. 

  Now we are going to study the character of the most general closed sets on the real line. 

  One of the most general proper closed subset of the real line is obtained by removing a countable disjoint class of open intervals. this process leads to some rather curious and complicated examples.

One of these examples is of particular importance. It was studied by Cantor and is usually called the Cantor Set.                                                           

                                                                               

Construction of Cantor Set :

                                     Let E₀ =[0,1].

  Let   E₁   be the subset of E obtained by removing the middle one third open interval (1/3,2/3) i.e. 

                                             E₁   = [0,1/3] ⋃[2/3,1]

  Let E₂ be the subset of  by removing the middle one third of the segments  (1/9,2/9 ) of [0,1/3] and (7/9,8/9 ) of [2/3,1] i.e.

                                             E₂ =[0,1/9]⋃ [2/9,1/3]⋃ [2/3,7/9]⋃ [8/9,1]

Continuing this process (of removing the middle one third of intervals)we have a sequence{E_n} suchthat 

               i . E_n ⊇ E_n+1  for all nꜪ N

               ii. each E is the union of intervals 2ⁿ each of length 3^-n

         The set E= ⋂E_n  is called Cantor Set.

PROPERTIES OF CANTOR SET :

 1. Cantor Set is non-empty :

                                               Since the Cantor Set is the intersection of decreasing sequence                                                 of closed 

                                                sets and each closed interval which has the first element 0, 0                                                     lies in the Cantor Set E.  Hence the Cantor Set is non-empty.

2. The Cantor Set is compact :

                                                We have every k-cell in  is compact. 

                                                ∴ for k=1, the 1-cell i.e. *= [0,1] is compact in R.

                                                           _{Since each * is a finite union of closed intervals , each * is closed.}

                                                          _{Also we have the Cantor Set E=⋂* , an arbitrary intersection of closed sets.}

                                                          ∴ E is closed.

                                                 Also since a closed subset of a compact set is compact and E is closed 

                                                subset of compact set * , the Cantor Set E is compact.

3. The Cantor Set contains no segments :

                                                Since the Cantor Set is obtained by taking intersection of closed sets* and 

                                                 each * has no segment , Cantor Set E has no segments.

4. The Cantor Set is perfect :

                                                We know that the Cantor Set E is closed.

                                                 To prove E is perfect , it remains to show that every point in E is a limit 

                                                 point of E.

                                                 Let x 𝟄 E and consider a neighborhood N_r(x)=(x-r, x+r ) of x.

                                                           Choose a positive integer n such that * <r.

                                                            Since x𝟄E implies x𝟄* for all n.

                                                            For some n𝟄N, we have x𝟄* .

                                                            Also since * is a union of * closed intervals *, we have x𝟄*.

                                                             There fore l(*)= * <r.

                                                             Let y 𝟄 *.

                                                             As x 𝟄 *, |x-y|<l(*) <r

                                                                             * |x-y|<r

                                                                             * y 𝟄 (x-r,x+r) = N_r(x)

                                                                              * y 𝟄  N_r(x)

                                                            Here a neighborhood of x contains a point y otherthan x.

                                                            Similarly every neighborhood of x contains a point of E other than X.

                                                            ∴ x is a limit point of E.

                                                            Since E is closed and each point of E is a limit point of E, the Cantor Set E                                                               is perfect.