I simply cannot wrap my head around the fact that the Cantor set has neither interior points nor isolated points.
As I understand it, an interior point $a\in A$ is one such that there is at least one neighborhood of $a$ that is a subset of $A$. Moreover, an isolated point $b\in B$ is one such that there is at least one neighborhood of $b$ that is disjoint of $B$.
How can the Cantor set fit both criterion?
If the Cantor set $C$ has no isolated points, then a point $c\in C$ must be a limit point. That is, every $\varepsilon$-neighborhood of $c$ intersects $C$ in some point different from $c$. Furthermore, if the Cantor set has no interior points, then every $\varepsilon$-neighborhood of $c\in C$ must not be a subset of $C$. Somehow, putting these definitions together does not seem to help me "see" this in my mind since it is so counter-intuitive.
Does anyone know of a more "natural" way to view this? Also, would $\mathbb{Q}\subset\mathbb{R}$ fit this criteria as well?