I have recently rediscovered (after several years) the wonder of the Cantor set (so rich and so beautiful!). I have two questions that are unrelated, but they are both about Cantor sets.
- Let $K$ be a non-empty compact, perfect, metric space such that $K \simeq K \times K$. Is $K$ necessarily homeomorphic to the Cantor set, or the Hilbert cube or some combination of both?
- Let $C$ be the Cantor set, $K$ the set of points $\exp(i2\pi x)$ where $x\in C $, and $S$ the set of all chords between points of $K$. Is $S$ convex?