Preference relation ($x⪰y$)

A weak preference relation records that one alternative is at least as desirable as another. Rational preferences are complete and transitive. Their strict preference and indifference relation are the asymmetric and symmetric parts. On a consumption space, continuity is expressed by a closed set of weakly ranked pairs.

This is the Fell topology restricted to closed graphs of complete, transitive, continuous preference relations. Both the upper limit condition and approximation of every limiting weak comparison are needed. A large limiting indifference class can make the second requirement stronger than convergence of numerical utility differences.

On $[0,\infty )$ the utility functions $u_n(x)=x/n$ converge uniformly on every bounded set to $u(x)=0$. Their preference relations remain the usual order, while the limiting utility function makes all pairs indifferent. The weak comparison $0⪰1$ in the limit cannot be approximated by comparisons from the varying graphs. Uniform convergence does guarantee the upper graph-limit condition, because convergent pairs remain in a fixed bounded set.

For a preorder, the indifference relation is an equivalence relation. Passing to its equivalence classes removes ties and produces a partial order.

The strict preference associated with a weak preference relation retains comparisons that do not hold in the reverse direction.