On the metric reflection of a pseudometric space in ZF

In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.

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Finding a measure, a ring theoretical approach

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v6i1.6864 ◽

2016 ◽

Vol 6 (1) ◽

pp. 1

Author(s):

Ali Parsian

Keyword(s):

Probability Theory ◽

Theoretical Approach ◽

Additive Function ◽

Characteristic Functions ◽

Uniform Probability ◽

Pseudometric Space

Let $S$ be a nonempty set and $F$ consists of all $Z_{2}$ characteristic functions defined on $S$. We are supposed to introduce a ring isomorphic to $(P(S),\triangle,\cap)$, whose set is $F$. Then, assuming a finitely additive function $m$ defined on $P(S)$, we change $P(S)$ to a pseudometric space $(P(S),d_{m})$ in which its pseudometric is defined by $m$. Among other things, we investigate the concepts of convergence and continuity in the induced pseudometric space. Moreover, a theorem on the measure of some kinds of elements in $(P(S),m)$ will be established. At the end, as an application in probability theory, the probability of some events in the space of permutations with uniform probability will be determined. Some illustrative examples are included to show the usefulness and applicability of results.

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Weak completeness of the Bourbaki quasi-uniformity

Applied General Topology ◽

10.4995/agt.2001.3018 ◽

2001 ◽

Vol 2 (1) ◽

pp. 101 ◽

Cited By ~ 3

Author(s):

M.A. Sánchez Granero

Keyword(s):

Uniform Space ◽

Cluster Point ◽

Compact Sets ◽

Weak Completeness ◽

Pseudometric Space

The concept of semicompleteness (weaker than half-completeness) is defined for the Bourbaki quasi-uniformity of the hyperspace of a quasi-uniform space. It is proved that the Bourbaki quasi-uniformity is semicomplete in the space of nonempty sets of a quasi-uniform space (X,U) if and only if each stable filter on (X,U*) has a cluster point in (X,U). As a consequence the space of nonempty sets of a quasi-pseudometric space is semicomplete if and only if the space itself is half-complete. It is also given a characterization of semicompleteness of the space of nonempty U*-compact sets of a quasi-uniform space (X,U) which extends the well known Zenor-Morita theorem.

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Invariant Graph Partition Comparison Measures

Symmetry ◽

10.3390/sym10100504 ◽

2018 ◽

Vol 10 (10) ◽

pp. 504 ◽

Cited By ~ 1

Author(s):

Fabian Ball ◽

Andreas Geyer-Schulz

Keyword(s):

Invariant Measures ◽

Automorphism Groups ◽

Structural Difference ◽

Equivalence Classes ◽

Classical Counterpart ◽

Symmetric Graphs ◽

Invariant Graph ◽

Partition Space ◽

Comparison Measures ◽

Pseudometric Space

Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived.

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Best Proximity Point Theorem in Quasi-Pseudometric Spaces

Abstract and Applied Analysis ◽

10.1155/2016/9784592 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8

Author(s):

Robert Plebaniak

Keyword(s):

Approximate Solution ◽

Point Theorem ◽

Best Proximity Point ◽

Sufficient Conditions ◽

Optimal Solution ◽

Pseudometric Space

In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the errorinf⁡{d(x,y):y∈T(x)}, and hence the existence of a consummate approximate solution to the equationT(X)=x.

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The δ-completion of a quasi-pseudometric space

Topology and its Applications ◽

10.1016/j.topol.2015.04.002 ◽

2015 ◽

Vol 189 ◽

pp. 65-77

Author(s):

Demco Kalusokoma Mukongo ◽

Olivier Olela Otafudu

Keyword(s):

Pseudometric Space

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Quasi-pseudometric properties of the Nikodym-Saks space

Applied General Topology ◽

10.4995/agt.2003.2029 ◽

2003 ◽

Vol 4 (2) ◽

pp. 243 ◽

Cited By ~ 2

Author(s):

Jesús Ferrer

Keyword(s):

Functional Analysis ◽

Metric Space ◽

Measure Theory ◽

Complete Metric Space ◽

Additive Measure ◽

Topological Properties ◽

Pseudometric Space

For a non-negative finite countably additive measure μ defined on the σ-field Σ of subsets of Ω, it is well known that a certain quotient of Σ can be turned into a complete metric space Σ (Ω), known as the Nikodym-Saks space, which yields such important results in Measure Theory and Functional Analysis as Vitali-Hahn-Saks and Nikodym's theorems. Here we study some topological properties of Σ (Ω) regarded as a quasi-pseudometric space.

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LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

Glasgow Mathematical Journal ◽

10.1017/s0017089514000512 ◽

2014 ◽

Vol 57 (3) ◽

pp. 591-632

Author(s):

BERNHARD KRÖN ◽

JÖRG LEHNERT ◽

NORBERT SEIFTER ◽

ELMAR TEUFL

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Complete Metric Space ◽

Nilpotent Groups ◽

General Concept ◽

Hausdorff Group ◽

Vertex Transitive ◽

Definition Of ◽

Transitive Graphs ◽

Pseudometric Space

AbstractWe define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

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A sequentially compact non-compact quasi-pseudometric space

Monatshefte für Mathematik ◽

10.1007/bf01471209 ◽

1983 ◽

Vol 96 (4) ◽

pp. 269-270 ◽

Cited By ~ 4

Author(s):

J. Ferrer ◽

V. Gregori

Keyword(s):

Pseudometric Space

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Some Results on Quasi-Uniform Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1976-005-6 ◽

1976 ◽

Vol 19 (1) ◽

pp. 39-51 ◽

Cited By ~ 1

Author(s):

Karen S. Carter ◽

T. L. Hicks

Keyword(s):

Topological Space ◽

Uniform Space ◽

Uniform Structure ◽

Uniform Spaces ◽

General Method ◽

Pseudometric Space

Constructions are made of a T1 space which does not have a T1 completion and of a quasi-uniform space which is complete, but not strongly complete. An example relating to a completion due to Popa is given. An alternate definition for Cauchy filter, called C-filter, is examined and a construction of a C-completion is given. We discuss quasi-pseudometrics over a Tikhonov semifield RΔ. Every topological space is quasi-pseudometrizable over a suitable RΔ. It is shown that if a quasi-pseudometric space over RΔ is complete, the corresponding quasi-uniform structure is C-complete. A general method for constructing compatible quasiuniform structures is given.

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