Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

Abstract

We generalize the notions of quietness and semisymmetry defined by Doitchinov (1991) and Deák (1991) and we study the role of these extended notions on the coincidence of some well-known quasi-uniform completeness. In particular, it is shown that the bicompletion coincides (up to quasi-uniformism) with the standard D-completion in quiet -weakly pair Cauchy bounded quasi-uniform spaces and it coincides with -half-completion defined by Romaguera and Sánchez-Granero (2002), in T 0 -weakly quiet quasiuniform spaces. definition of Cauchy net or Cauchy filter (often nets and filters lead to equivalent theories). The reason for the difficulty to develop a satisfactory theory of completeness for the class of all quasi-uniform spaces is due to the difficulty to approach the notion of Cauchy net (filter) in an appropriate way from the case of uniform spaces to the case of quasiuniform spaces. More precisely, since uniform spaces belong to the class of quasi-uniform spaces, according to Doitchinov [1], a notion of Cauchy net in any quasi-uniform space has to be defined in such a manner that this definition provides the properties that convergent nets are Cauchy, and it agrees with the usual definition for uniform spaces. Moreover, the suggested completion must be a monotone operator with respect to inclusion and give rise to the usual uniform completion in the uniform case.Fletcher and Lindgren in [4,5] give two equivalent theories of completeness, the pair completeness or bicompleteness. The first one uses pair of filters (Ᏺ,Ᏻ) for which Ᏺ ∪ Ᏻ is a filter base and the second is modelled upon the presentations of the completion of a uniform space given by Bourbaki [6]. It turns out that this concept coincides (for quasiuniform spaces) with that of double completeness introduced by Császár in [7]. The authors proved that the bicompletion has a very nice behavior: every T 0 quasi-uniform space can be quasi-uniformly embedded in a unique (i.e., up to quasi-uniformism) T 0 quasi-uniform space; for uniform spaces the construction coincides with the usual one. In [1], Doitchinov has introduced the notion of D-completeness and he proved that a certain category of quasi-uniform T 2 -spaces, the so-called quiet spaces, has a nice behavior: every quiet T 2 -space can be quasi-uniformly embedded in a unique D-complete quiet T 2 -space; for uniform spaces (which are always quiet) the construction coincides with the usual one. D-completeness is founded on a concept of a Cauchy net which can be done satisfying some most natural basic requirements. The Doitchinov completion provides another sophisticated completion that is in general larger than the bicompletion. (Consider for instance the set of the rationals equipped with the Sorgenfrey quasimetric. The space is bicomplete and only its Doitchinov completion yields the expected completion.)The definition of Cauchy net or filter in quasi-uniform space which has been proposed in bicompleteness and D-completeness gives a completing theory which satisfies the above requirements given by Doit...