All Questions

Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable? Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the ...

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...