On two questions concerning tilings

We prove that a tiling of the plane by topological disks is locally finite at most boundary points of tiles, confirming a conjecture by Valette. This comes by way of a much more general theorem on tilings of topological vector spaces. We also investigate a question raised by Klee as to whether or not there is a tiling of separable Hilbert space by bounded convex tiles. We present evidence to support the conjecture that the answer is negative.