scholarly journals Regular bases at non-isolated points and metrization theorems

2012 ◽  
Vol 49 (1) ◽  
pp. 91-105 ◽  
Author(s):  
Fucai Lin ◽  
Shou Lin ◽  
Heikki Junnila
Keyword(s):  
Affirmative Answer ◽  
Metrizable Space ◽  
Regular Space ◽  
Countable Base ◽  
Locally Finite ◽  
Finite Base ◽  
Isolated Points ◽  
Metrizable Spaces ◽  
Perfect Space

In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space X is a metrizable space, if and only if X is a regular space with a σ-locally finite base at non-isolated points, if and only if X is a perfect space with a regular base at non-isolated points, if and only if X is a β-space with a regular base at non-isolated points. In addition, we also discuss the relations between the spaces with a regular base at non-isolated points and some generalized metrizable spaces. Finally, we give an affirmative answer for a question posed by F. C. Lin and S. Lin in [7], which also shows that a space with a regular base at non-isolated points has a point-countable base.

scholarly journals Preservations of so-metrizable spaces

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 801-807 ◽  
Author(s):  
Shou Lin ◽  
Ying Ge
Keyword(s):  
Topological Group ◽  
Affirmative Answer ◽  
Metrizable Space ◽  
Regular Space ◽  
Locally Finite ◽  
Open Network ◽  
Covering Mappings ◽  
Metrizable Spaces ◽  
Closed Mappings ◽  
Closed Image

A space is called an so-metrizable space if it is a regular space with a ?-locally finite sequentially open network. This paper proves that so-metrizable spaces are preserved under perfect mappings and under closed sequence-covering mappings, which give an affirmative answer to a question on preservations of so-metrizable spaces under some closed mappings. Also, we prove that the closed image of an so-metrizable space is an so-metrizable space if it is a topological group.

scholarly journals The Metrizability of Spaces whose Diagonals have a Countable Base

1977 ◽  
Vol 20 (4) ◽  
pp. 513-514 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Metrizable Space ◽  
Countable Base ◽  
Neighborhood Base ◽  
Isolated Points

AbstractIt is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

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 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.

On a Theorem of Arhangel'skiĭ Concerning Lindelöf P-Spaces

1975 ◽  
Vol 27 (2) ◽  
pp. 459-468 ◽  
Author(s):  
R. E. Hodel
Keyword(s):  
Continuum Hypothesis ◽  
Affirmative Answer ◽  
Regular Space ◽  
Countable Base ◽  
Cardinal Function ◽  
Generalized Continuum ◽  
Remarkable Result

1. Introduction. In [4] Arhangel'skiĭ proved the remarkable result that every regular space which is hereditarily a Lindelöf p-space has a countable base. As a consequence of the main theorem in this paper, we obtain an analogue of Arhangel'skiĭs result, namely that every regular space which is hereditarily an ℵi-compact strong ∑-space has a countable net. Under the assumption of the generalized continuum hypothesis (GCH), the main theorem also yields an affirmative answer to Problem 2 in Arhangel'skiĭs paper.In § 3 we introduce and study a new cardinal function called the discreteness character of a space. The definition is based on a property first studied by Aquaro in [1], and for the class of T1 spaces it extends the concept of Kicompactness to higher cardinals.

scholarly journals A function space from a compact metrizable space to a dendrite with the hypo-graph topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Hanbiao Yang ◽  
Katsuro Sakai ◽  
Katsuhisa Koshino
Keyword(s):  
Finite Number ◽  
Function Space ◽  
Metrizable Space ◽  
Vietoris Topology ◽  
Hilbert Cube ◽  
Graph Topology ◽  
Closed Sets ◽  
Continuous Map ◽  
Compact Metrizable Space ◽  
Isolated Points

Abstract Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ | ƒ : X → Y is continuous} as the subspace of the hyperspace Cld(X × Y ) of nonempty closed sets in X × Y endowed with the Vietoris topology. Let be the closure of ↓v C(X, Y ) in Cld(X ×Y ). In this paper, we shall prove that the pair , ↓v C(X, Y )) is homeomorphic to (Q, c0), where Q = Iℕ is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.

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 An extension of the Kegel–Wielandt theorem to locally finite groups

1996 ◽  
Vol 38 (2) ◽  
pp. 171-176
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni ◽  
Yaroslav P. Sysak
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Arbitrary Group ◽  
Famous Theorem ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Finite Products ◽  
Open Question

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

The Number of Closed Subsets of a Topological Space

1978 ◽  
Vol 30 (02) ◽  
pp. 301-314 ◽  
Author(s):  
R. E. Hodel
Keyword(s):  
Topological Space ◽  
Basic Problem ◽  
Cardinal Number ◽  
Metrizable Space ◽  
Infinite Cardinal ◽  
Closed Sets ◽  
Complete Answer ◽  
Metrizable Spaces

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

Non-Metrizable Uniformities and Proximities on Metrizable Spaces

1973 ◽  
Vol 25 (5) ◽  
pp. 979-981
Author(s):  
P. L. Sharma
Keyword(s):  
Metrizable Space ◽  
Compact Metrizable Space ◽  
Metrizable Spaces

In the literature there exist examples of metrizable spaces admitting nonmetrizable uniformities (e.g., see [3, Problem C, p. 204]). In this paper, this phenomenon is presented more coherently by showing that every non-compact metrizable space admits at least one non-metrizable proximity and uncountably many non-metrizable uniformities. It is also proved that the finest compatible uniformity (proximity) on a non-compact non-semidiscrete space is always non-metrizable.

Sign in / Sign up

Export Citation Format

Share Document