Pixie dust on the number line. Invented by mathematician
Georg Cantor, the middle-third Cantor set is commonly used in textbooks to introduce
fractals and
self-similarity.
To construct a Cantor set, start with the interval [0,1] and throw away the middle third (1/3,2/3). You now have two pieces, [0,1/3] ∪ [2/3,1]. From each of these, throw away the middle third, leaving four pieces. From each of those, throw away the middle third. Etc...
What is left? Each step removes 1/3 of the points you have. After n steps, the total length of the intervals in the set is (2/3)^n. Since the Cantor set can be covered with intervals whose total length is arbitrarily small (let n go to infinity), it has measure zero, or zero length.
However, despite having no length, the Cantor set has an infinite number of points. The endpoints of every removed middle-third interval for example, are part of the Cantor set, and there are an infinite number of these.
Is there anything else in the Cantor set? Consider the numbers in [0,1] when expressed in base-3. Any number which does not contain the digit "1" in its base-3 representation is part of the Cantor set; any number which does contain "1" is not in the Cantor set. Therefore, numbers such as 1/4 (0.020202020202... in ternary) which will never be endpoints, are nonetheless in the Cantor set.
In fact, it is possible to map the Cantor set to the real numbers [0,1] in one-to-one correspondence! Note that when expressed in ternary, the Cantor set contains every sequence of the two digits 0 and 2. Now replace the 2's with 1's and consider those sequences as binary expansions...
A consequence of this is that the Cantor set is uncountable.