Statistical and Ideal Convergences in Topology

Abstract

The notion of convergence wins its own important part in the branch of Topology. Convergences in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, partially ordered sets (in short, posets), and fuzzy ordered sets (in short, fosets) develop significant chapters that attract the interest of many studies. In particular, statistical and ideal convergences play their own important role in all these areas. A lot of studies have been devoted to these special convergences, and many results have been proven. As a consequence, the necessity to produce and extend new results arises. Since there are many results on different kinds of convergences in different areas, we present a review paper on this research topic in order to collect the most essential results, which leads us to provide open questions for further investigation. More precisely, we present and gather definitions and results which have been proven for different kinds of convergences, mainly on statistical/ideal convergences, in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, posets, and fosets. Based on this presentation, we provide new open problems for further investigation on related topics. The study of these problems will create new “roads”, enriching the branch of convergences in the field of Topology.

The Equivalence of Two Modes of Order Convergence

“…Following the ideal in [8], Sun and Li [9] studied the B-topology on posets and found that the o-convergence in a poset P is topological if and only if the poset P is S * -doubly continuous, which demonstrates the equivalence between the o-convergence being topological and the S * -double continuity of a poset. Moreover, the ideal-o-convergence, a generalized form of o-convergence established via ideals, was defined in posets by Georgiou et al [10,11]. Also, the authors obtained that the ideal-o-convergence in a poset P is topological if and only if the poset P is S * -doubly continuous.…”