Take a line segment of length 1. We'll call it C1.

Now remove the middle third. Call what's left C2.

Now remove the middle third of each line segment in C2. Call what's left C3.

We can keep doing this, at each stage removing the middle third of each of the line segments in Cn to form Cn+1.

Construction of cantor set

Draw pictures of C4 and C5.

If we suppose that the end points of C1 are 0 and 1, then we can mark on the end points of the line segments for the later Cn too. For example, C2 has end points 0,
1
3

,
2
3

and 1 as shown below.

Cantor sets with labels

Draw C3 and label the end points, and label the end points on your pictures of C4 and C5.

We can keep removing middle thirds infinitely many times. The set of points left having done it infinitely many times is called the Cantor set.

Which of the following points are in the Cantor set?


1
3

,
4
9

,
3
81

,
4
81

.

Explain how you decided which belong and which don't.

 

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See also the problem Smaller and Smaller.