On topological spaces whose topology is induced by a binary relation

Topologies Generated by Nested Collections

“…Therefore, we are obliged to characterize preorderability of topologies in a totally independent way, without using the classical results on orderability. (See [9,17] for a further discussion on these important nuances and subtleties concerning preorderability vs. orderability of a given topology).…”

“…Remark 2.6 About the notation and nomenclature used in Definition 2.5, we point out that in some papers (see e.g., [9,14,17]) the left R-topology (respectively, the right R-topology) is called "upper topology" (respectively, "lower topology"). However, that nomenclature may cause confusion when compared to the traditional one in the domain theory (see [15]), where, if a partial order is fixed on a set X , then usually the upper topology is the least topology such that all the principal lower sets {y ∈ X : y x}, x ∈ X , are closed.…”

“…Finally, the topology τ R whose subbasis is {Rx} x∈X {x R} x∈X is called the twosided R-topology, to which we will refer in the sequel as the topology induced by the binary relation R. (See [17] for further information). Remark 2.6 About the notation and nomenclature used in Definition 2.5, we point out that in some papers (see e.g., [9,14,17]) the left R-topology (respectively, the right R-topology) is called "upper topology" (respectively, "lower topology").…”

“…In particular, preorderable topologies (among others) have been characterized in these new terms, as topologies induced by some suitable binary relation with additional properties. Remark 1.1 As a matter of fact, topologies induced in various ways by binary relations of certain kinds had already been studied in the specialized literature (see e.g., [1,17,19,21,[27][28][29][30][31][32]). A suitable introduction to this particular topic is [28].…”

“…On the other hand, in [1,17,[28][29][30][31][32], the topologies induced by binary relations defined on a given set have been considered and analyzed in depth. In particular, preorderable topologies (among others) have been characterized in these new terms, as topologies induced by some suitable binary relation with additional properties.…”