Real Analysis – Subsets of the Cantor Set

Question

Does anyone know of any interesting subsets of the Cantor set? When I first started thinking about this, I thought that since the Cantor set $C$ is the intersection of disjoint unions of closed sets, those particular closed sets would be subsets of $C$. Those sets, however, happen to be intervals! The Cantor set contains no isolated points nor (open) intervals, so then what subsets does it contain besides $\varnothing$ and itself?

Answer 1

Some could find the Cantor set minus a point an interesting subspace of the Cantor set. Regardless of which point is deleted, they are homeomorphic to each other. For concreteness denote $C_0=C\setminus\{0\}$. This space is characterized by the following

Theorem: A topological space is homeomorphic to $C_0$ if and only if it is separable, metrisable, zero-dimensional, locally compact, noncompact and has no isolated points.

Now that $C_0$ has been introduced, the nonempty open subsets of the Cantor set have the following nice characterization:

Theorem: Any nonempty compact open subset of $C$ is homeomorphic to $C$. Any noncompact open subset of $C$ is homeomorphic to $C_0$.

Furthermore, this property of having exactly two kinds of open subsets, of which the others are compact and the others noncompact, characterizes the Cantor set among compact metrisable spaces, see Schoenfeld A.H. and Gruenhage G., An Alternate Characterization of the Cantor Set. Proceedings of the American Mathematical Society 53 (1975), 235-236 (available online for free).