> > In what follows, everything (posets, lattices, topological spaces) is
> > finite.

> > 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).

> > Thus set families are a proper generalization of set families.

> In your first step, you say a family of
> sets "gives rise naturally to a lattice."
> But only certain families of sets (such
> as a power set) are examples of lattices.
> So your argument breaks down at its first
> link.

> regards, chip