Di-uniformities and Hutton uniformities

Di-extremities and totally bounded di-uniformities

“…When giving examples it will clearly suffice to give only δ satisfying the extremity conditions, since (DE) may then be used to define δ which will automatically satisfy the co-extremity conditions. This is also the case for dimetrics [14] and difunctional uniformities [17]. Only when one removes the symmmetry condition (DE) to produce a quasi di-extremity, it is absolutely necessary to consider both parts.…”

“…RDR . In this article, we will use the alternative characterization of difunctional uniformity which was given in [17]. Definition 2.10.…”

“…Definition 2.10. ( [17]) Let (S, S) be a texture. The subfamilies U ⊆ F RR and U ⊆ F RCR satisfy the following conditions:…”

“…Di-uniform texture spaces and totally bounded di-uniformities were introduced byÖzçag and Brown in [14,16]. Later, they also gave a point-free representation of direlational uniformities in [17] as well as they characterized Hutton uniformities [11] as di-uniformities on Hutton textures.…”

“…We say that B is a δ-neighborhood of A if and only if A B and C is a δ-neighborhood of D if and only if C D. For the details about di-extremities, the reader is referred to [19] and [20], and the details of proximity and quasi-proximity spaces can be seen from [9,12,13,18,19]. Di-Uniform Texture Spaces: Di-uniform texture spaces were introduced in [14] and later, a point-free representation of direlational uniformities, called difunctional uniformities were given in [17]. We omit the details and recall some fundamental definitions and results about difunctional uniformities.…”