Athanasios Andrikopoulos

The Quasimetrization Problem in the (Bi)topological Spaces

It is our main purpose in this paper to approach the quasi-pseudometrization problem in (bi)topological spaces in a way which generalizes all the well-known results on the subject naturally, and which is close to a “Bing-Nagata-Smirnov style” characterization of quasi-pseudometrizability.

On the construction of non-empty choice sets

In this article, we analyze the problem of choice based on potentially ill-behaved binary relations that may fail to possess maximal elements. The approach, which is based on Duggan (Soc Choice Welf 28:491–506, 2007), is to consider every maximal subrelation (resp. minimal superrelation) satisfying a regularity condition (acyclicity, consistency, or negative consistency) and to unify the maximal elements of all such relations. Based on this procedure, we present a characterization of the Smith set, the Duggan set, the Schwartz set, and the generalized stable sets solution of an arbitrary binary relation over non-finite sets. Schwartz’s and Van Deemen’s results for asymmetric binary relations stated in the finite case, are also extended to this more general framework. Finally, we give a set theoretical description of the Duggan set and show that the Schwartz set offers a refinement of the Duggan set, and the Schwartz and Duggan sets are nested between the union of generalized stable sets and the Smith set.

A Solution to the Completion Problem for Quasi-Pseudometric Spaces

The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.

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

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-com- pletion defined by Romaguera and Sánchez-Granero (2002), in T0-weakly quiet quasi-uniform spaces.