Closed convergence of preference relations

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.