3.4 *The Cantor Set 165Thus, C2 = [O, i] urn,~] u [~, ~] u [~, 1].
Continuing inductively, if Cn is the union of 2n disjoint closed intervals
of length 3 ~ , we define Cn+I to be the result of removing from Cn the open
middle thirds of these intervals, each of length 3 !+ 1 • For example,
C3 = [o, 2 ~] U [ 227 , i] U [~, J 7 ] u [ 287 , ~] u [~, ~~] U [;~, ~] U [;~, ;~] u [;*, l].
At each stage, to get Cn+I we remove the open middle thirds of the 2n
disjoint closed intervals comprising Cn, and Cn+I is the union of the resulting
2n+l disjoint closed intervals, each of length 3 n\ 1 • Notice that
C1 2 C2 2. .. 2 Cn 2 Cn+I 2 .. ..
We define the Cantor set to be
00
C = n Cn.
n=l
That is , C is what is left over in [O, 1] after removing successively all the "open
middle third" sets, as described above. · D
Theorem 3.4.2 The Cantor set is compact.
Proof. Exercise 1. •Theorem 3.4.3 The Cantor set contains no nonempty open interval.
Proof. Exercise 2. •THE CANTOR SET AND TERNARY DECIMALS
Let us consider what numbers belong to the Cantor set. The Cantor set is
clearly nonempty; for example, ~ E C. In fact,
Lemma 3.4.4 If a is an endpoint of one of the disjoint closed intervals com-
prising some Cn, then a EC.
Proof. Exercise 4. •To characterize the numbers that belong to the Cantor set C , we resort
to ternary (base-three) decimal-like representation of real numbers. While the
word "decimal" signals base-ten, any natural number b > 1 can be used to
represent real numbers in decimal-like form. For lack of a better term, we shall
refer to these expressions as base-b decimals.
Definition 3.4.5 (Base-b "Decimals") Let b be any natural number greater
than 1. Then a "base-b decimal" is any expression of the form
K.d1d2 · · · dndn+I · · · (base b) or
-K.d1d 2 · · · dndn+I · · · (base b)where K is a natural number and Vi EN, di E {O, 1, 2, · · · , b - l}.