Subsets of the Cantor set
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).
The Cantor set is a very interesting subset of the Cantor set.
It is a metric space which is:
- Compact;
- Nowhere dense as a subset of $[0,1]$;
- It has no isolated points;
- Polish (completely metrizable with a dense countable subset);
- Zero dimensional.
Not only that, it is also universal with respect to that every Hausdorff, compact, second countable and zero dimensional space is homeomorphic to the Cantor set.
Furthermore, every separable metric space which is zero dimensional is homeomorphic to a subset of the Cantor space. This includes, in particular, the irrationals (with the usual metric, also known as the Baire space).
From this fact we have that the uncountable closed subsets of the Cantor space are themselves Cantor spaces, we also have that any countable product of Cantor spaces is homeomorphic to the Cantor space, and we can partition every Cantor set into continuum many disjoint Cantor sets.
This to show you that "most" interesting subspaces of the Cantor set are themselves Cantor sets...