Cantor's dust
The Cantor ternary set consists of all real numbers in the unit interval that doesn't contain the digit 1 in their ternary decimal representation.
Explanation
The endpoints are given in base 3 notation.
03 0.013 0.023 0.13 0.23 0.213 0.223 13 Step 0 Step 1 Step 2 Step 3 Step 4 Step 5
Endpoints
The endpoints are never removed. Since each step removes a finite number of intervals and the number of steps is countable, the set of endpoints is countable.
Intervals
In step 1 we remove the numbers between 0.13 and 0.23. As 0.13 is the same as 0.0222…3 we are actually removing all those ternary decimals with a 1 in the first decimal place.
In step 2 we remove the numbers between 0.013 and 0.023 and the numbers between 0.213 and 0.223. As 0.013 is the same as 0.00222…3 and 0.213 is the same as 0.20222...3 we are actually removing all those ternary decimals with a 1 in the second decimal place.
In step 3 we remove the numbers with a 1 in the third decimal place, and so on.
The numbers that are left are those whose ternary decimal representation consist entirely of 0’s and 2’s.
Cardinality
If we replace the two's by one's in the remaining numbers, we get exactly all numbers of binary decimal representation. The set of these numbers is uncountable.
Thus, the Cantor set has the same cardinality as the interval [0,1] and so as the set of real numbers.
Conclusion
After we remove all intervals, the number of points remaining is no less than the number we started with. That is quite amazing.