Topology is, first and foremost, the study of space(s), and how they can be mapped one to one. Whether they are open/closed, bounded/unbounded, compact/complex.
As opposed to the subject studied in school, a topology is a structure imputed to a lattice (such as the Boolean lattice of sets ordered by inclusion).
To have a topology,therefore, one has to have a set of alements first, and that is what question was about.
Indeed, if space is empty, it cannot have a nontrivial topology.