In this talk we are going to describe Sz´az’s construction of some class of metric spaces. Most of all we will analyze topological properties of metric spaces obtained by using Sz´az’s construction. In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we will discuss the relation between Sz´az’s construction and the “linking construction”. A particular attention will be drawn to the “floor” metric, the analysis of which provides some interesting observations

Bugajewski, Dariusz

English

University of Dayton

University of DaytoneCommonsSummer Conference on Topology and ItsApplications Department of Mathematics6-2017On a Construction of Some Class of Metric SpacesDariusz BugajewskiAdam Mickiewicz University of Poznan, ddbb@amu.edu.plFollow this and additional works at: http://ecommons.udayton.edu/topology_confPart of the Geometry and Topology Commons, and the Special Functions CommonsThis Topology + Algebra and Analysis is brought to you for free and open access by the Department of Mathematics at eCommons. It has beenaccepted for inclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons. For more information,please contact frice1@udayton.edu, mschlangen1@udayton.edu.eCommons CitationBugajewski, Dariusz, "On a Construction of Some Class of Metric Spaces" (2017). Summer Conference on Topology and Its Applications.8.http://ecommons.udayton.edu/topology_conf/8Motivations Definition Examples Properties Another example ReferencesOn a construction of some classof metric spacesDariusz BugajewskiDepartment of Mathematics and Computer ScienceAdam Mickiewicz University in Poznan´, PolandDayton, June 27-30, 2017Motivations Definition Examples Properties Another example ReferencesMotivationsDefinitionExamplesPropertiesAnother exampleReferencesMotivations Definition Examples Properties Another example ReferencesDefinitionDefinition (Aronszajn–Panitchpakdi, 1956)We call a metric space hyperconvex, if each family of closed balls{B(xi , ri )}i∈J such thatd(xi , xj) 6 ri + rj for i , j ∈ Jhas a nonempty intersection.RemarkFor normed spaces, hyperconvexity means that each pairwiseintersecting family of closed balls has a nonempty intersection.Motivations Definition Examples Properties Another example ReferencesExamplesExamplesThe following spaces are hyperconvex:R;Rn with a “maximum” norm;l∞ and L∞(R);CR(K ), where K is a compact and extremally disconnectedHausdorff topological space.Motivations Definition Examples Properties Another example ReferencesExamples (continuation)ExamplesThe space R2 with the euclidean norm is not hyperconvex.However, the same space with the “maximum” norm is.Motivations Definition Examples Properties Another example ReferencesThe radial metricMotivations Definition Examples Properties Another example ReferencesThe ”river” metricMotivations Definition Examples Properties Another example ReferencesThe construction of certain metricsDefinition (pictorial; Borkowski, DB, Przybycien´)Motivations Definition Examples Properties Another example ReferencesLinking constructionDefinition (informal; Aksoy, Maurizi)Let (Xc , dc) and (Xi , di ) for i ∈ I be metric spaces. Assume thatfor every i ∈ I the intersection Xc ∩ Xi consists of exaclty onepoint - let us denote it by xi - and that for i , j ∈ I the intersectionXi ∩ Xj is empty or it contains only the point xi = xj .Let usdenote X := Xc ∪⋃i∈I Xi and let us define the functionϕ : X × X → [0,+∞) by formulaeψ(x , y) :=dc(x , y), if x , y ∈ Xcdi (x , y), if x , y ∈ Xidc(x , xi ) + di (xi , y), if x ∈ Xc i y ∈ Xidi (x , xi ) + dc(xi , xj) + dj(xj , y), if x ∈ Xi i y ∈ XjFactThe above defined function ψ is a metric on the set X .Motivations Definition Examples Properties Another example ReferencesLinking constructionDefinition (informal; Aksoy, Maurizi)Let (Xc , dc) and (Xi , di ) for i ∈ I be metric spaces. Assume thatfor every i ∈ I the intersection Xc ∩ Xi consists of exaclty onepoint - let us denote it by xi - and that for i , j ∈ I the intersectionXi ∩ Xj is empty or it contains only the point xi = xj . Let usdenote X := Xc ∪⋃i∈I Xi and let us define the functionϕ : X × X → [0,+∞) by formulaeψ(x , y) :=dc(x , y), if x , y ∈ Xcdi (x , y), if x , y ∈ Xidc(x , xi ) + di (xi , y), if x ∈ Xc i y ∈ Xidi (x , xi ) + dc(xi , xj) + dj(xj , y), if x ∈ Xi i y ∈ XjFactThe above defined function ψ is a metric on the set X .Motivations Definition Examples Properties Another example ReferencesMotivationsDefinitionExamplesPropertiesAnother exampleReferencesMotivations Definition Examples Properties Another example ReferencesThe definition of the metric ϕDefinition (Sza´z)Let (X , d) be a metric space, T : X → X be a mapping, g be ametric on T (X ) and let ∼ be such an equivalence relation on Xthat T is constant on its equivalence classes. Letϕ : X × X → [0,+∞) be defined by the formulaϕ(x , y) :=d(x , y), if x ∼ y ,d(x ,T (x)) + g(T (x),T (y)) + d(T (y), y),otherwise.FactIt is not difficult to check that the above defined function ϕ is ametric on the space X .Motivations Definition Examples Properties Another example ReferencesMotivationsDefinitionExamplesPropertiesAnother exampleReferencesMotivations Definition Examples Properties Another example ReferencesThe radial metric and the ”river” metricExampleLet X := R2, d = g be the euclidean metric, T ((x1, x2)) := (0, 0)and (x1, x2) ∼ (y1, y2)⇔ the points (0, 0), (x1, x2), (y1, y2) arecolinear.Then the metric ϕ is the well-known radial metric.ExampleLet X := R2, d = g be the euclidean metric, T ((x1, x2)) := (x1, 0)and (x1, x2) ∼ (y1, y2)⇔ x1 = y1.Then the metric ϕ is the well-known ”river” metric.Motivations Definition Examples Properties Another example ReferencesThe radial metric and the ”river” metricExampleLet X := R2, d = g be the euclidean metric, T ((x1, x2)) := (0, 0)and (x1, x2) ∼ (y1, y2)⇔ the points (0, 0), (x1, x2), (y1, y2) arecolinear.Then the metric ϕ is the well-known radial metric.ExampleLet X := R2, d = g be the euclidean metric, T ((x1, x2)) := (x1, 0)and (x1, x2) ∼ (y1, y2)⇔ x1 = y1.Then the metric ϕ is the well-known ”river” metric.Motivations Definition Examples Properties Another example ReferencesMotivationsDefinitionExamplesPropertiesAnother exampleReferencesMotivations Definition Examples Properties Another example ReferencesPropertiesFactIf d = g, the mapping T is continuous, the space (X , d) iscomplete and the equivlence classes of the relation ”∼” are closed,then the metric ϕ is complete.FactIf the metrics d and g are bounded, then the metric ϕ is alsobounded.FactIf the relation ”∼” is the equality relation, then every cluster pointof the space (X , ϕ) is a fixed point of the mapping T .Motivations Definition Examples Properties Another example ReferencesPropertiesFactIf d = g, the mapping T is continuous, the space (X , d) iscomplete and the equivlence classes of the relation ”∼” are closed,then the metric ϕ is complete.FactIf the metrics d and g are bounded, then the metric ϕ is alsobounded.FactIf the relation ”∼” is the equality relation, then every cluster pointof the space (X , ϕ) is a fixed point of the mapping T .Motivations Definition Examples Properties Another example ReferencesPropertiesFactIf d = g, the mapping T is continuous, the space (X , d) iscomplete and the equivlence classes of the relation ”∼” are closed,then the metric ϕ is complete.FactIf the metrics d and g are bounded, then the metric ϕ is alsobounded.FactIf the relation ”∼” is the equality relation, then every cluster pointof the space (X , ϕ) is a fixed point of the mapping T .Motivations Definition Examples Properties Another example ReferencesMotivationsDefinitionExamplesPropertiesAnother exampleReferencesMotivations Definition Examples Properties Another example ReferencesThe ”floor” metricExampleLet X := R, d = g be the euclidean metric, T (x) := bxc andx ∼ y ⇔ x = y . Then ϕ is a certain metric on the real line whichwe will denote by ϕb·c.FactThe metric ϕb·c is neither translation invariant nor homogeneous.It is not metrically convex, however it is complete.Motivations Definition Examples Properties Another example ReferencesThe ”floor” metricExampleLet X := R, d = g be the euclidean metric, T (x) := bxc andx ∼ y ⇔ x = y . Then ϕ is a certain metric on the real line whichwe will denote by ϕb·c.FactThe metric ϕb·c is neither translation invariant nor homogeneous.It is not metrically convex, however it is complete.Motivations Definition Examples Properties Another example ReferencesShape of ballsFactLet x ∈ Z and r > 0. ThenBϕb·c(x , r) =[x − brc, x − brc+ r − brc) ∪ [x − brc+ 1, x + r).FactLet x ∈ R and r > 0. ThenBϕb·c(x , r) = Bϕb·c(bxc, r − (x − bxc)) ∪ {x}.RemarkAnalogous formulae hold for closed balls.Motivations Definition Examples Properties Another example ReferencesShape of ballsFactLet x ∈ Z and r > 0. ThenBϕb·c(x , r) =[x − brc, x − brc+ r − brc) ∪ [x − brc+ 1, x + r).FactLet x ∈ R and r > 0. ThenBϕb·c(x , r) = Bϕb·c(bxc, r − (x − bxc)) ∪ {x}.RemarkAnalogous formulae hold for closed balls.Motivations Definition Examples Properties Another example ReferencesShape of ballsFactLet x ∈ Z and r > 0. ThenBϕb·c(x , r) =[x − brc, x − brc+ r − brc) ∪ [x − brc+ 1, x + r).FactLet x ∈ R and r > 0. ThenBϕb·c(x , r) = Bϕb·c(bxc, r − (x − bxc)) ∪ {x}.RemarkAnalogous formulae hold for closed balls.Motivations Definition Examples Properties Another example ReferencesShape of balls – examplesExamplesBϕb·c(1, 32)=[0, 12) ∪ [1, 52)0 121 2 52Bϕb·c(34 , 1)=[0, 14) ∪ {34}0 14341Motivations Definition Examples Properties Another example ReferencesShape of balls – examplesExamplesBϕb·c(1, 32)=[0, 12) ∪ [1, 52)0 121 2 52Bϕb·c(34 , 1)=[0, 14) ∪ {34}0 14341Motivations Definition Examples Properties Another example ReferencesForm of open setsFactA subset of the space (R, ϕb·c) is ϕ-open if and only if it is a unionof pairwise disjoint maximal intervals with the property that if theright endpoint of any of these intervals in an integer, then thisendpoint does not belong to that interval.CorollaryA ball in the space (R, ϕb·c) need not be a connected set. Forexample B(1, 32)=[0, 12) ∪ [1, 52) is the union of two open sets.Motivations Definition Examples Properties Another example ReferencesForm of open setsFactA subset of the space (R, ϕb·c) is ϕ-open if and only if it is a unionof pairwise disjoint maximal intervals with the property that if theright endpoint of any of these intervals in an integer, then thisendpoint does not belong to that interval.CorollaryA ball in the space (R, ϕb·c) need not be a connected set. Forexample B(1, 32)=[0, 12) ∪ [1, 52) is the union of two open sets.Motivations Definition Examples Properties Another example ReferencesMeasure of noncompactnessFactLet A ⊂ [k , k + 1) for some k ∈ Z. The Kuratowski measure ofnoncompactness of the set A (in view of the metric ϕb·c) expressesby the formulaα(A) = 2 max({x − k | x is a cluster point of the set A} ∪ {0}).RemarkIn the above formula a cluster point is considered in view of theeuclidean metric.Motivations Definition Examples Properties Another example ReferencesMeasure of noncompactnessFactLet A ⊂ [k , k + 1) for some k ∈ Z. The Kuratowski measure ofnoncompactness of the set A (in view of the metric ϕb·c) expressesby the formulaα(A) = 2 max({x − k | x is a cluster point of the set A} ∪ {0}).RemarkIn the above formula a cluster point is considered in view of theeuclidean metric.Motivations Definition Examples Properties Another example ReferencesMeasure of noncompactnessCorollaryLet A be a bounded subset of R. The Kuratowski measure ofnoncompactness of the set A (in view of the metric ϕb·c) expressesby the formulaα(A) = 2 maxk∈Zmax({x − k∣∣∣∣ x is a cluster pointof the set A ∩ [k, k + 1)}∪ {0})Motivations Definition Examples Properties Another example ReferencesMotivationsDefinitionExamplesPropertiesAnother exampleReferencesMotivations Definition Examples Properties Another example ReferencesReferencesA. G. Aksoy and B. MauriziMetric trees, hyperconvex hulls and extensionsTurk J. Math. 32 (2008), 219–234M. Borkowski, D. Bugajewski and A. BurchardtOn topological properties of metrics defined via generalized ”linkingconstruction”, submittedM. Borkowski, D. Bugajewski, H. Przybycien´Hyperconvex spaces revisitedBull. Austral. Math. Soc., 68 (2003), 191–203D. Bugajewski, E. GrzelaczykA fixed point theorem for hyperconvex spacesArch. Math. (Basel), 75 (2000), 395–400D. Bugajewski, E. GrzelaczykOn the measures of noncompactness in some metric spacesNew Zealand J. Math., 27(2) (1998), 177–182Motivations Definition Examples Properties Another example ReferencesReferencesA´. Sza´zA common generalization of the postman, radial, and river metricsRostock. Math. Kolloq. 67 (2012), 89–125Motivations Definition Examples Properties Another example ReferencesThank youfor your attention.

On a Construction of Some Class of Metric Spaces

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On a Construction of Some Class of Metric Spaces

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