Cantor Set

Cantor Set

The (“middle third”) Cantor set is constructed as follows: 0. Set .

1. Remove the middle third (as an open interval) of this set, leaving .
2. From each of closed intervals from remove the open middle third to give a new set that consists of closed intervals. Note that consists of closed intervals, each of length (so their total length is , as ). Now the set is the (middle third) Cantor set.

Prop Since each is closed, is closed (as it is the intersection of closed sets). Prop C is non-empty: it contains the endpoints of every interval that we remove (in fact it contains uncountably many points, which we will prove later).