On the scale of a uniform space

The scale of a quasi-uniform space

“…We define our scale of a quasi-uniform space motivated by the definition of the Hausdorff hyperspace quasi-uniformity of a quasi-uniform space as introduced in [3]. In Proposition 5.1.3 we show that if the original space is a uniform space then our scale is the same as the scale investigated by Bushaw and Kent, see [6] and [16]. In Corollary 5.6.1 we also show that the quasi-uniform space given by a Hausdorff hyperspace quasi-uniformity is quasi-uniformly embedded into the left-sided scale of a quasi-uniform space.…”

“…In the last section, we shall summarize pseudometrizabilty, separability, the Lindelof condition, and the first and second axioms of countability of the scale of a uniform space. A large portion of this chapter was obtained from [6] and [16]. In order to avoid repetitions it will sometimes happen that we state a result in this chapter but delay its proof until Chapter 5.…”

“…Szaz in [38] The following definition is according to Kent (see [16] In the following, we are going to define a uniformity on the scale of a uniform space.…”

“…It will be constructed with a simple method that has been used by Kent in [16] by introducing an equivalence relation R on P.…”

“…In 1967 D. C. Kent published a paper [16] defining an partial order in the retracted scale which makes it a complete distributive lattice and the canonical mapping from the scale to the retracted scale is order preserving. The lattice operations in the retracted scale are uniformly continuous.…”