scholarly journals Monad Metrizable Space

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1891
Author(s):  
Orhan Göçür
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Real Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Metrizable Spaces

Do the topologies of each dimension have to be same and metrizable for metricization of any space? I show that this is not necessary with monad metrizable spaces. For example, a monad metrizable space may have got any indiscrete topologies, discrete topologies, different metric spaces, or any topological spaces in each different dimension. I compute the distance in real space between such topologies. First, the passing points between different topologies is defined and then a monad metric is defined. Then I provide definitions and some properties about monad metrizable spaces and PAS metric spaces. I show that any PAS metric space is also a monad metrizable space. Moreover, some properties and some examples about them are presented.

scholarly journals so-metrizable spaces and images of metric spaces

2021 ◽  
Vol 19 (1) ◽  
pp. 1145-1152
Author(s):  
Songlin Yang ◽  
Xun Ge
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Metrizable Space ◽  
Open Mapping ◽  
Generalized Metric Space ◽  
Locally Finite ◽  
Generalized Metric Spaces ◽  
Covering Mapping ◽  
Generalized Metric ◽  
Metrizable Spaces

Abstract so-metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so-metrizable space if it has a σ \sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so-metrizable space if and only if it is an so-open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so-open, σ \sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.

scholarly journals τ-metrizable spaces

2018 ◽  
Vol 19 (2) ◽  
pp. 253
Author(s):  
A.C. Megaritis
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Cardinal Number ◽  
Metrizable Space ◽  
Cardinal Numbers ◽  
Metrizable Spaces

<p>In [1], A. A. Borubaev introduced the concept of τ-metric space, where τ is an arbitrary cardinal number. The class of τ-metric spaces as τ runs through the cardinal numbers contains all ordinary metric spaces (for τ = 1) and thus these spaces are a generalization of metric spaces. In this paper the notion of τ-metrizable space is considered.</p>

scholarly journals Peano compactifications and propertySmetric spaces

1980 ◽  
Vol 3 (4) ◽  
pp. 695-700
Author(s):  
R. F. Dickman
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Metrizable Space ◽  
Connected Sets ◽  
Metrizable Spaces ◽  
Necessary And Sufficient ◽  
Separable Metric Space

Let(X,d)denote a locally connected, connected separable metric space. We say theXisS-metrizable provided there is a topologically equivalent metricρonXsuch that(X,ρ)has PropertyS, i.e. for anyϵ>0,Xis the union of finitely many connected sets ofρ-diameter less thanϵ. It is well-known thatS-metrizable spaces are locally connected and that ifρis a PropertySmetric forX, then the usual metric completion(X˜,ρ˜)of(X,ρ)is a compact, locally connected, connected metric space, i.e.(X˜,ρ˜)is a Peano compactification of(X,ρ). There are easily constructed examples of locally connected connected metric spaces which fail to beS-metrizable, however the author does not know of a non-S-metrizable space(X,d)which has a Peano compactification. In this paper we conjecture that: If(P,ρ)a Peano compactification of(X,ρ|X),Xmust beS-metrizable. Several (new) necessary and sufficient for a space to beS-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.

scholarly journals Characterizations of sn-metrizable spaces

2003 ◽  
Vol 74 (88) ◽  
pp. 121-128 ◽  
Author(s):  
Ying Ge
Keyword(s):  
Metric Space ◽  
Metrizable Space ◽  
Finite Point ◽  
Locally Finite ◽  
Metrizable Spaces

We give some characterizations of sn-metrizable spaces. We prove that a space is an sn-metrizable space if and only if it has a locally-finite point-star sn-network. As an application of the result, a space is an sn-metrizable space if and only if it is a sequentially quotient, ? (compact), ?-image of a metric space.

scholarly journals Fixed points for pairs of mappings indcomplete topological spaces

1993 ◽  
Vol 16 (2) ◽  
pp. 259-266 ◽  
Author(s):  
Troy L. Hicks ◽  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Large Class ◽  
Distance Function ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Triangular Inequality

Several important metric space fixed point theorems are proved for a large class of non-metric spaces. In some cases the metric space proofs need only minor changes. This is surprising since the distance function used need not be symmetric and need not satisfy the triangular inequality.

scholarly journals Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

10.29007/pw5g ◽  
2018 ◽  
Author(s):  
Larry Moss ◽  
Jayampathy Ratnayake ◽  
Robert Rose
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Fractal Sets ◽  
Cauchy Completion ◽  
Sweeping Generalization ◽  
Initial Algebra ◽  
Finite Metric Spaces ◽  
Set Of Points

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra ofa certain functor on the category of bipointed sets. Leinster 2011 offersa sweeping generalization of this result. He is able to represent many of what would be intuitivelycalled "self-similar" spaces using (a) bimodules (also called profunctors or distributors),(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction offinal coalgebras for the types of functors of interest using a notion of resolution. In addition to thecharacterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.Our major contribution is to suggest that in many cases of interest, point (c) above on resolutionsis not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces ofinterest as the Cauchy completion of an initial algebra,and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.This generalizes Hutchinson's 1981 characterization of fractal attractors asclosures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space.of dyadic rational numbers in [0,1].Our second contribution is not completed at this time, but it is a set of results on \emph{metric space}characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,and our interest in quotient metrics comes from their paper. So in terms of (a)--(c) above, our workdevelops (a) and (b) in metric settings while dropping (c).

QUANTIFIED INTUITIONISTIC LOGIC OVER METRIZABLE SPACES

2019 ◽  
Vol 12 (3) ◽  
pp. 405-425
Author(s):  
PHILIP KREMER
Keyword(s):  
Intuitionistic Logic ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Constant Domain ◽  
Current Article ◽  
Image Position ◽  
Metrizable Spaces ◽  
Rational Line

AbstractIn the topological semantics, quantified intuitionistic logic, QH, is known to be strongly complete not only for the class of all topological spaces but also for some particular topological spaces — for example, for the irrational line, ${\Bbb P}$, and for the rational line, ${\Bbb Q}$, in each case with a constant countable domain for the quantifiers. Each of ${\Bbb P}$ and ${\Bbb Q}$ is a separable zero-dimensional dense-in-itself metrizable space. The main result of the current article generalizes these known results: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality ≤ the space’s weight; consequently, QH is strongly complete for any separable zero-dimensional dense-in-itself metrizable space with a constant countable domain. We also prove a result that follows from earlier work of Moerdijk: if we allow varying domains for the quantifiers, then QH is strongly complete for any dense-in-itself metrizable space with countable domains.

scholarly journals Computing of Existence and Uniqueness Solutions for Differential Equations in Metric Spaces by Using MATLAB

2022 ◽  
Vol 10 (01) ◽  
Author(s):  
Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Topological Properties ◽  
Computer Solution ◽  
Matlab Program ◽  
Proposed Model ◽  
Is Design

A metric space is a set along with a measurement on the set, A metric actuates topological properties like open and shut sets, which lead to the investigation of more theoretical topological spaces. It also has many applications in functional analysis. The aim of this work is design and develop highly efficient algorithms that provide the existence of unique solutions to the differential equation in metric spaces using MATLAB. The quality algorithm was used and developed to solve the differential equation in metric spaces. For accurate results. The proposed model contributed to providing an integrated computer solution for all stages of the solution starting from the stage of solving differential equations in metric space and the stage of displaying and representing the results graphically in the MATLAB program

scholarly journals Metrizability of multiset topological spaces

2021 ◽  
Vol 13(62) (2) ◽  
pp. 683-696
Author(s):  
Karishma Shravan ◽  
Binod Chandra Tripathy
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Metrizable Spaces

In this paper, we have investigated one of the basic topological properties, called Metrizability in multiset topological space. Metrizable spaces are those topological spaces which are homeomorphic to a metric space. So, we first give the notion of metric between two multi-points in a finite multiset and studied some significant properties of a multiset metric space. The notion of metrizability is then studied by using this metric. Besides, the Urysohn’s lemma which is considered to be one of the important tools in studying some metrization theorems in topology is also discussed in context with multisets.

On Metrizability of Topological Spaces

1968 ◽  
Vol 20 ◽  
pp. 795-804 ◽  
Author(s):  
Carlos J. R. Borges
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moore Space ◽  
Necessary And Sufficient ◽  
Preceding Question

Our present work is divided into three sections. In §2 we study the metrizability of spaces with a Gδ-diagonal (see Definition 2.1). In §3 we study the metrization of topological spaces by means of collections of (not necessarily continuous) real-valued functions on a topological space. Our efforts, in §§2 and 3, are directed toward answering the following question: “Is every normal, metacompact (see Definition 2.4) Moore space a metrizable space?” which still remains unsolved. (However, Theorems 2.12 through 2.15 and Theorem 3.1 may be helpful in answering the preceding question.) In §4 we prove an apparently new necessary and sufficient condition for the metrizability of the Stone-Čech compactification of a metrizable space and hence for the compactness of a metric space.

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