Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classicalmereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected (Varzi in Noûs 31:26–58, 1997). In this paper we urge that this problem can be met with a para...