scholarly journals Topologies induced by metrics with disconnected range

1982 ◽  
Vol 25 (1) ◽  
pp. 133-142
Author(s):  
Kevin Broughan
Keyword(s):  
Metric Space ◽  
Metrizable Space ◽  
Uniform Limit ◽  
Irrational Numbers ◽  
Compatible Metrics ◽  
Completely Metrizable

In a metric space (X, d) a ball B(x, ε) is separated if d(B(x, ε), X\B(x, ε)] > 0. If the separated balls form a sub-base for the d-topology then Ind X = 0. The metric is gap-like at x if dx(X) is not dense in any neighbourhood of 0 in [0, ∞). The usual metric on the irrational numbers, P, is the uniform limit of compatible metrics (dn), each dn being gap-like on P. In a completely metrizable space X if each dense Gδ is an Fσ then Ind X = 0.

Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

1992 ◽  
Vol 35 (4) ◽  
pp. 439-448 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Approximation Theory ◽  
Metric Space ◽  
Polish Space ◽  
Complete Metric Space ◽  
Approximation Of Functions ◽  
Lower Envelopes ◽  
Constructive Approximation ◽  
Hausdorff Approximation ◽  
Set Of Points ◽  
Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

Remetrization in Strongly Countabledimensional Spaces

1969 ◽  
Vol 21 ◽  
pp. 748-750 ◽  
Author(s):  
B. R. Wenner
Keyword(s):  
Metric Space ◽  
Metrizable Space ◽  
Dimension Function ◽  
Image Position ◽  
Necessary And Sufficient

Although the Lebesgue dimension function is topologically invariant, the dimension-theoretic properties of a metric space can sometimes be made clearer by the introduction of a new, topologically equivalent metric. A considerable amount of effort has been devoted to the problem of constructing such metrics; one example of the fruits of this research is the following theorem by Nagata (2, Theorem 5).In order that dim R ≦ n for a metrizable space R it is necessary and sufficient to be able to define a metric p(x, y) agreeing with the topology of R such that for every ∊ > 0 and for every point x oƒ R,implyA metric ρ which satisfies the condition of this theorem is called Nagata's metric (this term was introduced, to the best of the author's knowledge, by Nagami (1, Definition 9.3)).

Approximation Par des Fonctions Lipschitziennes et Critere de Convergence Etroite D'une Suite de Probabilites

1984 ◽  
Vol 27 (4) ◽  
pp. 514-516 ◽  
Author(s):  
I. Rihaoui
Keyword(s):  
Weak Convergence ◽  
Metric Space ◽  
Bounded Function ◽  
Uniform Limit ◽  
Bounded Functions ◽  
New Criterion ◽  
Uniformly Continuous

AbstractIn this paper, we prove that a real valued bounded function, defined on a metric space and uniformly continuous is the uniform limit of a sequence of Lipschitzian bounded functions.As a consequence, a new criterion for the weak convergence of probabilities is given.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

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 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 Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

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 Quasi-uniform and uniform convergence of Riemann and Riemann-type integrable functions with values in a Banach space

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1899-1913
Author(s):  
Pratikshan Mondal ◽  
Lakshmi Dey ◽  
Ali Jaker
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Uniform Convergence ◽  
Sequence Space ◽  
Metric Space ◽  
Uniform Limit ◽  
Bounded Interval ◽  
Integrable Functions ◽  
Different Types ◽  
Sequence Of Functions

In this article, we study quasi-uniform and uniform convergence of nets and sequences of different types of functions defined on a topological space, in particular, on a closed bounded interval of R, with values in a metric space and in some cases in a Banach space. We show that boundedness and continuity are inherited to the quasi-uniform limit, and integrability is inherited to the uniform limit of a net of functions. Given a sequence of functions, we construct functions with values in a sequence space and consequently we infer some important properties of such functions. Finally, we study convergence of partially equi-regulated* nets of functions which is shown to be a generalized notion of exhaustiveness.

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