Let \({\mathcal {X}}\) be a locally compact topological space. We consider two spaces of continuous preference relations on \({\mathcal {X}}\) . A local continuous quasiorder is a continuous, complete, quasitransitive binary relation defined on a closed subset of \({\mathcal {X}}\) . Let \(Q({\mathcal {X}})\) be the set of all such relations. A local continuous strict partial order is a continuous partial order (with no indifference) defined on a closed subset of \({\mathcal {X}}\) . Let \(P({\mathcal {X}})\) be the set of all such relations. There is a natural bijection between \(Q({\mathcal {X}})\) and \(P({\mathcal {X}})\) . We endow both sets with the Fell topology, and show that under mild conditions, they are compact Hausdorff spaces, compact metrizable spaces, continua, or even contractible Peano continua. If \({\mathcal {X}}\) is a compact metric space, then duplex multiutility representations define continuous functions into \(Q({\mathcal {X}})\) and \(P({\mathcal {X}})\) from a space of compact collections of utility functions, endowed with its own Fell topology. Furthermore, any continuous function \(\phi :{\mathcal {X}}{{\longrightarrow }}{\mathcal {Y}}\) induces continuous functions \(\phi ^@:Q({\mathcal {X}}){{\longrightarrow }}Q({\mathcal {Y}})\) and \(\phi ^\P :P({\mathcal {X}}){{\longrightarrow }}P({\mathcal {Y}})\) . We thus obtain two endofunctors Q and P on the category of compact Hausdorff spaces, which are naturally isomorphic to each other. Finally, we show that these endofunctors are “continuous”: if \({\mathcal {X}}\) is the limit of a chain \({\mathcal {X}}_1 \longleftarrow {\mathcal {X}}_2 \longleftarrow {\mathcal {X}}_3 \longleftarrow \cdots \) of compact Hausdorff spaces, then \(Q({\mathcal {X}})\) is the limit of the corresponding chain \(Q({\mathcal {X}}_1) \longleftarrow Q({\mathcal {X}}_2) \longleftarrow Q({\mathcal {X}}_3) \longleftarrow \cdots \) (and likewise for P).
