Stripes on rectangular tilings

We consider a class of cut-and-project sets $\Lambda = \Lambda_F \times \zahl$intheplane\mathrm{.}Let$L=\Lambda+w\real$,$w\in\real^2$, be a countable union of parallel lines. Then either (1) $L$ is a discrete family of lines, (2) $L$ is a dense subset of ${\mathrm{\Re }}^2$, or (3) each connected component of the closure of $L$ is homeomorphic to $[0,1]\times \mathrm{\Re }$.