As we also know, the normal Cantor set is constructed by keeping cutting out the middle 1/3 set.
So, can we construct a Cantor set $E$ by keeping cutting out the nth 1/m set, where $1 \leq n \leq m$, $m >2$ are integers. It means that we can not only cut the middle set, but also the first set or the last set when $m=3$.
Moreover, $E$ will keep the properties that the normal Cantor set has.
(i) $E$ is uncountable.
(ii) $E$ has zero measure.
(iii) $E$ is nowhere dense.
The above there properties are easy to prove. Do I lose any important property that $E$ may have?