Uniform approximation of topological spaces
The authors consider the upper topology of compact \(T_2\)-ordered spaces (in the sense of Nachbin). It is well known that these topologies are the \(T_0\)-analogue of compact \(T_2\)-spaces. They call such a space \(X\) uniformly approximated if its canonical quasi-uniformity (that is the one determining the associated compact \(T_2\)-ordered space) has a base consisting of hypergraphs of continuous functions \(f:X\rightarrow X.\) (For a function \(f:X\rightarrow Y\) between topological spaces the hypergraph of \(f\) is defined by \(\{(x,y):f(x)\in \overline{\{y\}}\}.\)) They show that uniformly approximated spaces carry with them functional means of approximating points, opens and compacts and provide various characterizations of these spaces by means of topology, uniform topology, order theory and locale theory. For instance, they show that the specialization order on each uniformly approximated space yields an \(FS\)-domain and that each \(FS\)-domain equipped with its Scott-topology is uniformly approximated; furthermore, these two translations are inverses of each other. They also prove that the spectrum of a finitary proximity lattice is a uniformly approximated space and that every uniformly approximated space arises as the spectrum of some finitary proximity lattice.
scientific article; zbMATH DE number 1308050
scientific article