# Bounded components of a set

Hi everyone: Suppose $E$ is a closed set with empty interior in $\mathbb{R}^{k}$, $k\geq2$. Can $\mathbb{R}^{k}\setminus E$ have infinitely many bounded components?

Take $k=2$ and $E=\text{Grid}$, where $\text{Grid}:=\{(x,y) \mid x \text{ or }y \text{ is an integer}\}$.