This method is used, for example, in the proof that there is no free complete lattice.

The Scott-open sets form a complete lattice when ordered by inclusion.

In fact, free complete lattices do generally not exist.

The free complete lattice on three generators does not exist; it is properly a proper class.

There are various other mathematical concepts that can be used to represent complete lattices.

Then the set of fixed points of f in L is also a complete lattice.

It remains to be proven that P is complete lattice.

Consider a partially ordered set (P, ) that is a complete lattice.

Some authors work even with more general structures than the real line, like complete lattices.

Within this complete lattice, the compact elements are exactly the finite sets.