A family of sets gives rise naturally to a lattice. A lattice gives
rise naturally to a poset. A poset gives rise naturally to a
topological space. But a topological space is just a certain type of
set family, and thus set families generalize topological spaces.

All these generalizations are proper: not every lattice would induce a
set family, nor would every poset induce a lattice, nor would every
topological space induce a poset (only the T0 spaces).