Preorderable topologies and order-representability of topological spaces
A total preorder is called continuous (resp. semicontinuous) with respect to a given topology provided that the order topology (resp. lower or upper topology) induced by the total preorder is coarser than the given topology. Topologies that coincide with the order topology (resp. with the lower topology) induced by a total preorder are called preorderable (resp. lower preorderable). The authors characterize such topologies. They also say that a topology defined on a nonempty set satisfies the continuous representability property (CRP) if every continuous total preorder defined on the given set admits a numerical representation by means of a continuous real-valued order-monomorphism. A topology defined on a nonempty set satisfies the semicontinuous representability property (SRP) if every lower (upper) semicontinuous total preorder defined on the given set admits a numerical representation by means of a lower (upper) semi-continuous real-valued order-monomorphism. Given a topological space \((X,\tau)\) the continuous (semicontinuous) representability property CRP (SRP) on \((X,\tau)\) is characterized by means of the satisfaction of the second countability axiom by every preorderable (lower preorderable) topology on \(X\) that is coarser than \(\tau\).
orderability properties on topological spaces
order-representability properties on topological spaces
scientific article; zbMATH DE number 5625493
scientific article