  The quest to realize topological band structures in artificial matter is
strongly focused on lattice systems, and only quantum Hall physics is known to
appear naturally also in the continuum. In this letter, we present a proposal
based on a two-dimensional cloud of atoms dressed to Rydberg states, where
excitations propagate by dipolar exchange interaction, while the Rydberg
blockade phenomenon naturally gives rise to a characteristic length scale,
suppressing the hopping on short distances. Then, the system becomes
independent of the atoms' spatial arrangement and can be described by a
continuum model. We demonstrate the appearance of a topological band structure
in the continuum characterized by a Chern number $C=2$ and show that edge
states appear at interfaces tunable by the atomic density.
