All Questions

Is there a subset $X$ of plane with two points $x, y$ such that each one of $X \setminus \{x\}$, $X \setminus \{y\}$ is isometric to $X$? I tried hard to construct a counterexample but failed. Sorry ...

Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...