A set created by taking the interval (0,1), and removing the middle third (exclusively), then removing the middle third of the two remaining pieces, and continuing this process ad infinitum.
It's uncountable, among other things.
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.

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