Editing Cantor set (section)

== Composition ==
Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., [[Lebesgue measure|measure]]) of the unit interval remaining can be found by total length removed.  This total is the [[geometric progression]]

:<math>\sum_{n=0}^\infty \frac{2^n}{3^{n+1}} = \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \cdots =  \frac{1}{3}\left(\frac{1}{1-\frac{2}{3}}\right) = 1.</math>

So that the proportion left is  <math>1 - 1 = 0</math>.

This calculation suggests that the Cantor set cannot contain any interval of non-zero length. It may seem surprising that there should be anything left—after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing [[open set]]s (sets that do not include their endpoints). So removing the line segment <math display="inline">\left(\frac{1}{3}, \frac{2}{3}\right)</math> from the original interval <math>[0, 1]</math> leaves behind the points {{sfrac|1|3}} and {{sfrac|2|3}}. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining.  So the Cantor set is not [[empty set|empty]], and in fact contains an [[uncountably infinite]] number of points (as follows from the above description in terms of paths in an infinite binary tree).

It may appear that ''only'' the endpoints of the construction segments are left, but that is not the case either. The number {{sfrac|1|4}}, for example, has the unique ternary form 0.020202... = {{overline|0.|02}}. It is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on. Since it is never in one of the middle segments, it is never removed. Yet it is also not an endpoint of any middle segment, because it is not a multiple of any power of {{sfrac|1|3}}.<ref name="College">{{citation
 | last1 = Belcastro | first1 = Sarah-Marie
 | last2 = Green | first2 = Michael
 | date = January 2001
 | doi = 10.2307/2687224
 | issue = 1
 | journal = The College Mathematics Journal
 | page = 55
 | title = The Cantor set contains <math>\tfrac{1}{4}</math>? Really?
 | volume = 32| jstor = 2687224
 }}</ref>
All endpoints of segments are ''terminating'' ternary fractions and are contained in the set
:<math> \left\{x \in [0,1] \mid \exists i \in \N_0: x \, 3^i \in \Z \right\} \qquad \Bigl(\subset \N_0 \, 3^{-\N_0} \Bigr) </math>
which is a [[countably infinite]] set.
As to [[cardinality]], [[almost all]] elements of the Cantor set are not endpoints of intervals, nor [[rational number|rational]] points like {{sfrac|1|4}}.  The whole Cantor set is in fact not countable.