This note studies the relationship between Hutton's quasi-uniformities and Shi's quasi-uniformities. It is shown that when $L$ satisfies``multiple choice principle" for co-prime elements, the category of Hutton's quasi-uniform spaces is a bireflective full subcategory of the category of Shi's quasi-uniform spaces. Especially, if the remote-neighborhood mapping defined by Shi preserves arbitrary...

If the symmetry condition in the definition of a pseudometric is deleted, the notion of a quasi-pseudometric is obtained. Asymmetric distance functions already occur in the work of Hausdorff in the beginning of the twentieth century when in his book on set-theory [27] he discusses what is now called the Hausdorff metric of a metric space. A family of pseudo-metrics on a set generates a uniformi...

The aim of this note is to show how various facts in classical topology connected with semicontinuous functions and the semicontinuous quasi-uniformity have their counterparts in pointfree topology. In particular, we introduce the localic semicontinuous quasi-uniformity, which generalizes the semicontinuous quasiuniformity of a topological space (known to be one of the most important examples o...

and Applied Analysis 3 The quasi-uniform space X,U will always be considered to be a topological space with the topology obtained by using as the family of all neighborhoods of a point x0 ∈ X all sets of the formU x0 {x ∈ X : x0, x ∈ U}, whereU runs overU. Such a topology is called topology of the quasi-uniformity. Note that each topology onX is induced by a quasi-uniformity see 8, 9 , the most...

Algorithmic random number generators require recurring sequences with very long periods and good multivariate uniformity properties. Point sets and sequences for quasi-Monte Carlo numerical integration need similar multivariate uniformity properties as well. It then comes as no surprise that both types of applications share common (or similar) construction methods. However, there are some diffe...