Quotient-reflective and bireflective subcategories of the category of preordered sets

Local pre-Hausdorff extended pseudo-quasi-semi metric spaces

“…P reT 2 at p and P reT 0 2 at p could be equivalent. For example, for the category Top of topological spaces, by Theorem 2 as well as for the category Preord of preordered (sets with re ‡exive and transitive relations on them) sets and monotone (relation preserving) maps, by Theorems 6.3 and 6.4 of [11], P reT 2 at p and P reT 0 2 at p are equivalent. Note, also, that all objects of a set-based arbitrary topological category may be P reT 2 at p. For example, it is shown, in [15], that all Cauchy spaces [12] are P reT 2 at p.…”

T₁ Approach Spaces

“…In 1991, Baran [2] introduced local T 1 separation property in order to de…ne the notion of strong closedness [2] in set-based topological category which forms closure operators in sense of Dikranjan and Giuli [14,15] in some well known topological categories Conv (category of convergence spaces and continuous maps) [6,18,23], Prord (category of preordered sets and order preserving maps) [7,15] and SUConv (category of semiuniform convergence spaces and uniformly continuous maps) [9,24]. Furthermore, Baran [2] generalized T 1 axiom of topology to topological category which is used to de…ne regular, completely regular and normal objects [4,5] in topological categories.…”

T0 and T1 semiuniform convergence spaces

“…These notions are used in [2,8,12,13] to generalize each of the notions of compactness, connectedness, Hausdorffness, and perfectness to arbitrary set-based topological categories. Moreover, it is shown, in [11,12,14] that they form appropriate closure operators in the sense of Dikranjan and Giuli [20] in some well-known topological categories.…”