  We show that spatial curvature has a significant effect on the existence of
entanglement islands in cosmology. We consider a homogeneous, isotropic
universe with thermal radiation purified by a reference spacetime. Arbitrarily
small positive curvature guarantees that the entire universe is an island.
Proper subsets of the time-symmetric slice of a closed or open universe can be
islands, but only if the cosmological constant is negative and sufficiently
large in magnitude.
