Definition 5.2.12: Cantor Middle Third Set

Start with the unit interval

S₀ = [0, 1]

Remove from that set the middle third and set

S₁ = S₀ \ (1/3, 2/3)

Remove from that set the two middle thirds and set

S₂ = S₁ \ { (1/9, 2/9) (7/9, 8/9) }

Continue in this fashion, where

S_n+1 = S_n \ { middle thirds of subintervals of S_n }

Then the Cantor set C is defined as

C = S_n

For a list of properties of the Cantor set, look at the theory of compact and perfect sets.

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