Topology and its Applications最新文献

英文中文

Metric duality for Abelian groups

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109155

Piotr Niemiec

The main aim of the paper is to introduce the concept of metric duality in the category of topological Abelian groups that extends the classical notion of duality for normed vector spaces and behaves quite nicely for LCA groups (equipped with nice metrics). In particular, it is shown that each Polish LCA group admits a reflexive proper metric and, more generally, all LCA groups possess reflexive (proper) metric structures.

引用次数: 0

Editorial on the Mary Ellen Rudin Young Researcher Award competition 2023

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109152

Alan Dow, Jan van Mill (Editors-in-Chief Topology and its Application)

引用次数: 0

Some properties involving feeble compactness, III: (Weakly) compact-bounded topological groups

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109149

J.A. Martínez-Cadena, Á. Tamariz-Mascarúa

We study two topological properties weaker than feeble compactness in the class of (para)topological groups, the compact-boundedness and weak compact-boundedness, both introduced by Angoa, Ortiz-Castillo and Tamariz-Mascarúa in [2]. First, given a subgroup H of a topological group G, we show how to extend these properties from the quotient space

G / H

to G; this, in the cases when H is a compact, locally compact or (weakly) compact-bounded subgroup. Secondly, we prove the main result of this article: if a Tychonoff space X is compact-bounded and not scattered, then the free topological group

F (X)

and the free Abelian topological group

A (X)

admit a non-trivial metrizable quotient group; thus extending Theorem 4.7 by Leiderman and Tkachenko in [15]. Finally, we study the r-weakly compact-bounded subsets of a topological space X. We show that r-weak compact-boundedness is a productive property. Moreover, sufficient conditions are given in order for a C-compact subset of a paratopological group G to become an r-weakly compact-bounded subset. This article is part of a larger work developed in [16] and [17].

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引用次数: 0

Anti-absorbing ternary operations on metric spaces

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109153

Leonid V. Kovalev

The existence of a median-type ternary operation on a metric space is known to have a number of implications for the geometry of the space. For such operations, if two of the three arguments coincide, they also coincide with the output of the operation. We consider ternary operations with the opposite property: if two of the arguments coincide, the output is equal to the third one. The existence of such an operation is a necessary condition for the space to be an absolute retract.

引用次数: 0

Discrete Morse theory on ΩS2

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109185

Lacey Johnson , Kevin Knudson

One classical consequence of Morse theory is that it provides a description of a cell structure for a CW-complex having the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's

F^{+} K

construction to model the loop space of the sphere

S^{2}

, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of

Ω S^{2}

引用次数: 0

The hyperspace of k-dimensional closed convex sets

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109154

Adriana Escobedo-Bustamante , Natalia Jonard-Pérez

For every

n \geq 2

, let

K_{k}^{n}

denote the hyperspace of all k-dimensional closed convex subsets of the Euclidean space

R^{n}

endowed with the Atouch-Wets topology. Let

K_{k, b}^{n}

be the subset of

K_{k}^{n}

consisting of all k-dimensional compact convex subsets. In this paper we explore the topology of

K_{k}^{n}

and

K_{k, b}^{n}

and the relation of these hyperspaces with the Grassmann manifold

G_{k} (n)

. We prove that both

K_{k}^{n}

and

K_{k, b}^{n}

are Hilbert cube manifolds with a fiber bundle structure over

G_{k} (n)

. We also show that the fiber of

K_{k, b}^{n}

with respect to this fiber bundle structure is homeomorphic with

R^{\frac{k (k + 1) + 2 n}{2}} \times Q

, where Q stands for the Hilbert cube.

引用次数: 0

The first-countability in generalizations of topological groups with ideal convergence

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109150

Xin Liu, Shou Lin, Xiangeng Zhou

The study of convergence in topological groups has become a frontier research subject. In many cases, the first-countability is an important and strong condition. Based on

I_{s n}

-continuity, the present paper discusses how topology and algebra are related through a notion of continuity generated by ideal convergence. We introduce the classes of generalizations of topological groups, give the structures of

I_{s n}

-topological groups by certain sequential coreflections, and obtain generalized metric properties of

I

-snf-countable para-

I_{s n}

-topological groups.

Let

I

be an admissible ideal on the set

N

of natural numbers. The following results are obtained.

(1)
Every T₂, $I$ -snf-countable para- $I_{s n}$ -topological group is an sn-quasi-metrizable and cs-submetrizable space.
(2)
A T₀, $I_{s n}$ -topological group is an $I$ -snf-countable space if and only if it is a cs-metrizable space satisfying that each sequentially open subset is $I_{s n}$ -open.

These show the unique role of

I_{s n}

-continuity in the study of topological groups and related structures, and present a version of topological algebra using the notion of ideals.

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引用次数: 0

On MM-ω-balancedness and FR(Fm)-factorizable semi(para)topological groups

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109183

Liang-Xue Peng, Yu-Ming Deng

In the second part of this article, we introduce a notion which is called MM-ω-balancedness in the class of semitopological groups. We show that if G is a semitopological (paratopological) group, then G is topologically isomorphic to a subgroup of the product of a family of metacompact Moore semitopological (paratopological) groups if and only if G is regular MM-ω-balanced and

I r (G) \leq ω

. If G is a

T_{0}

bM-ω-balanced semitopological group and

f : G \to H

is an open continuous homomorphism of G onto a first-countable semitopological group H such that

\ker (f)

is a countably compact subgroup of G, then H is a metacompact developable space.

In the third part of this article, we introduce notions of

F R

-factorizability and

F m

-factorizability. We give some equivalent conditions that a semitopological (paratopological) group is

F R

-factorizable or

F m

-factorizable. If G is a Tychonoff

F R

(

F m

)-factorizable semitopological group and

f : G \to H

is a continuous open homomorphism of G onto a semitopological group H, then H is

F R

(

F m

)-factorizable. If G is a

F R

(

F m

)-factorizable paratopological group and

f : G \to H

is a continuous d-open homomorphism of G onto a paratopological group H, then H is

F R

(

F m

)-factorizable.

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引用次数: 0

A computational framework for weighted simplicial homology

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109177

Andrei C. Bura , Neelav S. Dutta , Thomas J.X. Li , Christian M. Reidys

We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring

R = F [[π]]

. This is achieved by starting from a basis for classical homology of the n-th skeleton for the underlying complex with coefficients in the residue field

F

and then lifting it to a basis for the weighted homology with coefficients in the ring R. Using the latter, a bijection is established between

n + 1

and n dimensional simplices whose weight ratios provide the exponents of the π-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over R. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of R, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently.

{"title":"A computational framework for weighted simplicial homology","authors":"Andrei C. Bura , Neelav S. Dutta , Thomas J.X. Li , Christian M. Reidys","doi":"10.1016/j.topol.2024.109177","DOIUrl":"10.1016/j.topol.2024.109177","url":null,"abstract":"<div><div>We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring <math><mi>R</mi><mo>=</mo><mi>F</mi><mo>[</mo><mo>[</mo><mi>π</mi><mo>]</mo><mo>]</mo></math>. This is achieved by starting from a basis for classical homology of the n-th skeleton for the underlying complex with coefficients in the residue field <math><mi>F</mi></math> and then lifting it to a basis for the weighted homology with coefficients in the ring R. Using the latter, a bijection is established between <math><mi>n</mi><mo>+</mo><mn>1</mn></math> and n dimensional simplices whose weight ratios provide the exponents of the π-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over R. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of R, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109177"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Metrizable spaces homeomorphic to the hyperspace of nonblockers of singletons of a continuum

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109151

David Maya, Fernando Orozco-Zitli, Emiliano Rodríguez-Anaya

A continuum is a nondegenerate compact connected metric space. The hyperspace of all nonempty closed subsets of a continuum X topologized by the Hausdorff metric is denoted by

2^{X}

. Given a continuum X, the subspace

NB (F_{1} (X))

2^{X}

consists of all elements

A \in 2^{X} - {X}

such that for each

x \in X - A

, the union of all subcontinua of X containing x and contained in

X - A

is a dense subset of X. The members of

NB (F_{1} (X))

are called nonblocker subsets of the singletons of the continuum X. In this paper, we show that each proper nonempty open subset U of a compact metric space can be embedded in a continuum X such that U and the hyperspace of nonblocker subsets of X are homeomorphic. This answers a question posed by J. Camargo, F. Capulín, E. Castañeda-Alvarado and D. Maya.

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