2019/05/22 by Hanna Ćmiel, Franz‐Viktor Kuhlmann, Ćmiel, Hanna +3
Decision Sciences · Mathematics · #03E75 #06A05 #06A06 #06B23 #06B99 #06F20 #12J15 #12J20 #13A18 #47H09 #47H10 #54A05 #54C10 #54C60 #54E50 (Secondary) #54H25 (Primary) #Advanced Topology and Set Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Fuzzy and Soft Set Theory #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.1905.09930
openalex publication_date 2019/05/22 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
With a simple generic approach, we develop a classification that encodes and\nmeasures the strength of completeness (or compactness) properties in various\ntypes of spaces and ordered structures. The approach also allows us to encode\nnotions of functions being contractive in these spaces and structures. As a\nsample of possible applications we discuss metric spaces, ultrametric spaces,\nordered groups and fields, topological spaces, partially ordered sets, and\nlattices. We describe several notions of completeness in these spaces and\nstructures and determine their respective strengths. In order to illustrate\nsome consequences of the levels of strength, we give examples of generic fixed\npoint theorems which then can be specialized to theorems in various\napplications which work with contracting functions and some completeness\nproperty of the underlying space.\n Ball spaces are nonempty sets of nonempty subsets of a given set. They are\ncalled spherically complete if every chain of balls has a nonempty\nintersection. This is all that is needed for the encoding of completeness\nnotions. We discuss operations on the sets of balls to determine when they lead\nto larger sets of balls; if so, then the properties of the so obtained new ball\nspaces are determined. The operations can lead to increased level of strength,\nor to ball spaces of newly constructed structures, such as products. Further,\nthe general framework makes it possible to transfer concepts and approaches\nfrom one application to the other; as examples we discuss theorems analogous to\nthe Knaster--Tarski Fixed Point Theorem for lattices and theorems analogous to\nthe Tychonoff Theorem for topological spaces.\n