A textural view of the distinction between uniformities and quasi-uniformities

Di-uniformities and Hutton uniformities

“…General references on ditopological texture spaces include [1,2,3,4,5,6] and constant reference will be made to [13,14,15,16,17] for definitions and results relating to di-uniformities. Our standart references for quasi uniformity are [7,8,9].…”

“…Since dual covers are not well known concepts, we now recall from [1,15] some definitions and properties. However the dual covering quasi-uniformity and the equivalence with the diagonal quasi-uniformity were studied widely in [15].…”

“…Di-uniformity on a texture was first defined in [13] by giving descriptions in terms of direlations, dicovers and dimetrics and the concepts of completeness and total boundedness were introduced in [14]. The effect of a complementation and the relation with quasi-uniformity and uniformity were discussed in [15]. In this context the work [15] pointed out that di-uniformities provide a more unified setting for the study of quasiuniformity and uniformity than does the classical approach.…”

“…The effect of a complementation and the relation with quasi-uniformity and uniformity were discussed in [15]. In this context the work [15] pointed out that di-uniformities provide a more unified setting for the study of quasiuniformity and uniformity than does the classical approach.…”

“…It should be noted, that the symmetry condition (5) which guarantees a base of symmetric direlations for the direlational uniformity is quite different from the notion of symmetry for relations. In [15] an important result was obtained that a direlational uniformity on the discrete texture (X, P(X)) corresponds not to uniformity but to quasi uniformity. When the symmetry condition (5) is removed we obtain a direlational quasi-uniform texture space (S, S, U q ) [17].…”