Applied General Topology - Vol 27, No 1 (2026)

Tabla de contenidos

Finite lattices and large inductive dimension
Space of quasicontinuous functions with the topology of uniform convergence on semi-compacta
Ti-reflections and the Hartman-Mycielski construction in semitopological and paratopological groups
Fixed and periodic points of general mappings with applications
Simple closed curves contained in ε-boundaries of planar sets
An adaptation of the Vietoris topology for ordered compact sets
On local semirings induced by topologies: An algebraic approach to the Collatz conjecture
Lelek-like fans: endpoint-dense continua supporting topologically mixing maps
Strong convergence of an inertial iterative method for generalized nonexpansive mappings in Banach spaces
Simple polynomial equations over (2x2)-matrices
Some remarks on selectively highly divergent spaces
I-Fréchet-Urysohn property in Cα(X)
Subring structures in C(X) defined by absolute values
Best proximity points in graphical vector metric spaces and its consequences
Collective compactness, equicompactness and the degree of nondensifiability
Mappings contracting transverse axis of hyperbola
More on T-closed sets
Chaos on the interval in terms of distributional functions
On w-unicoherence, top-irreducibility and Whitney levels near the top
Various notions of topological transitivity in non-autonomous discrete dynamical systems
Remarks on fixed point assertions in digital topology, 11
On the lattice of e-open sets and od-compact spaces
Fixed point analysis of fuzzy graphical ?-contractions in non-Archimedean graphical fuzzy metric spaces with applications to nonlinear fractional-order differential models
I-Fréchet-Urysohn property in Cα(X)
Lelek-like fans: endpoint-dense continua supporting topologically mixing maps

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Access status: Abierto ,
Lelek-like fans: endpoint-dense continua supporting topologically mixing maps
(Universitat Politècnica de València, 2025-03-16) Banić, Iztok; Erceg, Goran; Jelić, Ivan; Kennedy, Judy; Slovenian Research Agency
[EN] The Lelek fan is the only smooth fan that has a dense set of end-points. In this paper, we study non-smooth fans with this property; i.e., we construct an uncountable family of pairwise non-homeomorphic such fans. Furthermore, we prove that each of them admits a topologically mixing non-invertible mapping as well as a topologically mixing homeomorphism.
Access status: Abierto ,
I-Fréchet-Urysohn property in Cα(X)
(Universitat Politècnica de València, 2026-03-16) Zhou, Xiangeng; National Natural Science Foundation of China; Natural Science Foundation of Fujian Province
[EN] In this paper, we introduce I-Fr´echet-Urysohn, strongly I-Fr´echet-Urysohn and strictly I-Fr´echet-Urysohn spaces,discuses their properties of countable tightness and mappings that preserve these spaces.Meanwhile, we discuss the internal characterizations of these spaces in C?(X).The following main theorem is obtained. Theorem. Let ? be a network of X. The following are equivalent. (1) Cα(X) is a strictly I-Fr´echet-Urysohn space. (2) Cα(X) is a strongly I-Fr´echet-Urysohn space. (3) Cα(X) is an I-Fr´echet-Urysohn space. (4) Every open α-cover of X contains an I-α-sequence. (5) If {Un}n∈N is a sequence of open α-cover of X, then there is an I-α-sequence {un}n∈N of X such that each un ∈ Un. (6) Cωα (X) is a strictly I-Fréchet-Urysohn space.
Access status: Abierto ,
Fixed point analysis of fuzzy graphical ψ-contractions in non-Archimedean graphical fuzzy metric spaces with applications to nonlinear fractional-order differential models
(Universitat Politècnica de València, 2026-05-07) Wong, Koon Sang; Salleh, Zabidin; Dhananjay, Gopal; University of Malaysia, Terengganu
[EN] Within graphical fuzzy metric spaces, the present study addresses a formulation for fuzzy graphical ψ-contractions as a novel family of contractions. Various fixed point results pertaining to the proposed contractions are established, supported by illustrative examples. Additionally, a discussion of the obtained findings extend and refine existing fixed point theorems is provided. To demonstrate their applicability, the study includes an application on establishing solution attainability for fractional-order nonlinear differential equations.
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On the lattice of e-open sets and od-compact spaces
(Universitat Politècnica de València, 2026-05-07) Taherifar, Ali
[EN] In this paper, we investigate od-compact spaces, a natural generalization of e-compact spaces (hence compact spaces), and study their interplay with the lattice-theoretic structure of e-open and open dense subsets, denoted by E(X) and OD(X), respectively. We first provide an algebraic characterization of od-compactness and e-compactness in terms of essential and Baer ideals of C(X), respectively. We then observe the relationships between od-compactness, connectedness, hyperconnectedness, and compactness, and examine the behavior of these notions under continuous open maps and products. Furthermore, we analyze the lattice properties of E(X) and OD(X), showing that they are closely related to the lattice of Baer and essential ideals of C(X), respectively. Several structural properties of these lattices, such as compactness, connectedness, and co-normality, are discussed, along with their preservation under homeomorphisms and ring isomorphisms. Finally, we establish isomorphisms between these lattices and their counterparts in the Stone-?ech compactification.
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Remarks on fixed point assertions in digital topology, 11
(Universitat Politècnica de València, 2026-04-30) Boxer, Laurence
[EN] The topic of fixed points in digital metric spaces has drawn yet more publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We discuss publications with bad assertions concerning fixed points of self-functions on digital images, as in some of our previous papers.
Access status: Abierto ,
Simple closed curves contained in ε-boundaries of planar sets
(Universitat Politècnica de València, 2025-12-03) Patrakeev, Mikhail; Volkov, Aleksei; Ministry of Science and Higher Education of the Russian Federation
[EN] The ε-boundary of a set A ⊆ R2 is the set { p ∈ R2 : ρ(p,A) = ε } , where ρ is the Euclidean distance. We prove that if A,B ⊆ R2 are nonempty, connected sets, A is bounded, and 0< ε < ρ(A,B), then the ε-boundary of A contains a simple closed curve (aka a Jordan curve) that separates A and B. This statement follows from the theorem which says that if ε>0 and A ⊆ R2 is a nonempty, bounded, connected set, then the boundary of each component of { p ∈ R2 : ρ(p,A) > ε } is a simple closed curve. Another corollary of this theorem is that the ε-boundary of a nonempty, bounded, connected set A ⊆ R2 contains a simple closed curve bounding the domain that contains the open ε-neighbourhood of A. In all these statements the connectivity condition can be significantly weakened. We also show that, for all ε>0, the ε-boundary of a nonempty, bounded set A ⊆ R2 contains a simple closed curve.
Access status: Abierto ,
Ti-reflections and the Hartman-Mycielski construction in semitopological and paratopological groups
(Universitat Politècnica de València, 2025-12-03) Reyes, Boris
[EN] We show that, given a semitopological group G, the semitopological groups Ti (G?) and (Ti(G))· , for each i ? {0,1,2} , and (Gsr)· and (G·)sr, are topologically isomorphic. Moreover, we prove that, given a paratopological group G, the paratopological groups Reg(G·) and (Reg(G))· are topologically isomorphic.
Access status: Abierto ,
Mappings contracting transverse axis of hyperbola
(Universitat Politècnica de València, 2025-12-03) Roy, Kushal
[EN] In this paper a new contractive type mapping known as mapping contracting transverse axis of hyperbola is introduced. This mapping can reduce the length of transverse axis of a hyperbola. This is a geometric technique that is connected to the study of geometric characteristics of certain curves defined over metric spaces. The paper is decorated by some suitable examples that support our proven results and showing the distinctness of our mapping from the usual contractive type mappings. Our proposed mapping also admits discontinuity at fixed point, thus gives a new solution to an open problem posed by B.E. Rhoades. Finally a geometric figure is illustrated to describe the speciality of our mapping.
Access status: Abierto ,
Best proximity points in graphical vector metric spaces and its consequences
(Universitat Politècnica de València, 2025-12-03) Fallahi, Kamal; Ardakani, Halimeh; Karimizad, Seyedeh Sara
[EN] Regarding the vector metric space introduced by Cevik and Altun, we first introduce the concept of a graphical vector metric space. We then define several topological notions in this setting, including Dedekind complete Riesz spaces, vectorially G-continuous mappings, vectorially G-proximal mappings, and the vector P-property. Subsequently, we establish several best proximity point theorems in these spaces using the newly introduced concepts. Various consequences and an illustrative example are also provided to support our definitions and results. One of the distinguishing features of this work, compared to related research, is that it encompasses previously introduced results by considering comparable and suitably close elements.
Access status: Abierto ,
Space of quasicontinuous functions with the topology of uniform convergence on semi-compacta
(Universitat Politècnica de València, 2025-12-03) Barman, Neelim Kumar; Hazarika, Debajit
[EN] We introduce a new set-open topology on function spaces namely the semi-compact-open topology. The topology of uniform convergence on semi-compacta lies between the topology of pointwise convergence and the topology of uniform convergence. We show that for the space of quasicontinuous functions the topology of uniform convergence on semi-compacta coincides with the semi-compact-open topology. We also investigate results relating to induced functions,  cardinal invariants and Ascoli like properties on the space of quasicontinuous functions when equipped with the topology of uniform convergence on semi-compacta.
Access status: Abierto ,
Finite lattices and large inductive dimension
(Universitat Politècnica de València, 2025-12-03) Georgiou, Dimitrios N.; Hattori, Yasunao; Megaritis, Athanasios; Sereti, Fotini
[EN] The Ordered Set Theory is a branch of Mathematics that studies partially ordered sets (usually posets) and lattices. The meaning of dimension is one of the main parts of this field. In particular, the covering dimension, the Krull dimension and the small inductive dimension have been studied extensively for the class of finite lattices. In this paper,we insert a new meaning of dimension for finite lattices called large inductive dimension. We study various of its properties based on minimal covers. Also, given two finite lattices, we study the dimension Ind of their linear sum, Cartesian, lexicographic and rectangular product, investigating the "behavior" of this dimension. In addition, we study relations of this new dimension with the small inductive dimension, covering dimension and Krull dimension, presenting various facts andexamples that strengthen the corresponding results.
Access status: Abierto ,
Fixed and periodic points of general mappings with applications
(Universitat Politècnica de València, 2026-03-31) Bisht, Ravindra K; Bhatt, Arvind; Joshi, Radha; Pant, R. P.; Rakocevic, Vladimir
[EN] In this paper, we establish new results on the existence, uniqueness, and convergence of general mappings in two cases: mappings that admit only a fixed point, and mappings that admit both fixed and periodic points. Our findings apply to both contractive and non-expansive mappings, thereby broadening their scope of applicability. We demonstrate that the established fixed-point theorems have significant number-theoretic consequences. The analysis of these classes of mappings leads directly to a generating equation for Pythagorean triples and, ultimately, to the classical negative Pell equation U2 - 2Y2 = -1.
Access status: Abierto ,
An adaptation of the Vietoris topology for ordered compact sets
(Universitat Politècnica de València, 2026-03-09) Caruvana, Christopher; Holshouser, Jared
[EN] We introduce a natural topology on powers of a space that is inspired by the Vietoris topology on compact subsets. We then place this topology in context with other product topologies; specifically, we compare this topology with the Tychonoff product, the box product, and Bell's uniform box topology. We identify a variety of topological properties for the specific case when the ground space is discrete. When the ground space is the Euclidean real line, we show that the resulting power is not Lindelöf, and hence, not Menger. This shows that, unlike the the Vietoris topology on unordered compact subsets, covering properties of the ground space need not transfer to the Vietoris power.
Access status: Abierto ,
Simple polynomial equations over (2x2)-matrices
(Universitat Politècnica de València, 2025-12-03) Chatyrko, Vitalij A.; Karassev, Alexandre; Natural Sciences and Engineering Research Council of Canada
[EN] We consider the polynomial equation Xn + an-1 ⋅ Xn-1 + ⋯ + a1 ⋅ X + a0 ⋅ I = O over (2 x 2)-matrices X with the real entries, where I is the identity matrix, O is the null matrix, ai ∈ ℝ for each i and n ≥ 2 . We discuss its solution set S supplied with the natural Euclidean topology. We completely describe S. We also show that dim S =2.
Access status: Abierto ,
Chaos on the interval in terms of distributional functions
(Universitat Politècnica de València, 2026-03-09) Balibrea, Francisco; Rucká, Lenka
[EN] In paper [2], Balibrea and Smítal studied distributional functions of a continuous map f of the interval I and they proposed to use the size" of the set of all distributional functions F (f) (with respect to different pairs of points) as a measure of chaos of the map f . We show that not the size of F (f) itself, but the existence of a distributional function which is strictly increasing, can be used to detect chaotic behavior of the interval map f . The main result shows that any topologically transitive unimodal map of the interval generates strictly increasing distributional function. On the other hand, the existence of a strictly increasing distributional function for f implies positive topological entropy of f , which proves the conjecture from [5] for interval maps.
Access status: Abierto ,
Strong convergence of an inertial iterative method for generalized nonexpansive mappings in Banach spaces
(Universitat Politècnica de València, 2026-02-09) Patel, Prashant; Shukla, Rahul
[EN] We introduce an iterative technique with an inertial term that converges strongly to a fixed point of mappings satisfying Condition (E). Our results extend existing work by providing a robust numerical method for solving fixed point problems in Banach spaces. To demonstrate the effectiveness of our approach, we present numerical examples of a mapping that is not nonexpansive but satisfies Condition (E). Furthermore, we illustrate the convergence behaviour of our algorithm for different choices of initial guesses and coefficients, using MATLAB to validate the theoretical results. This work contributes to the broader framework of fixed point theory and offers practical insights for solving nonlinear problems in applied mathematics.
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Some remarks on selectively highly divergent spaces
(Universitat Politècnica de València, 2026-02-24) Tuyen, Luong Quoc; Truc, Nguyen Xuan; Tuyen, Tuyen
[EN] In this paper, we study selectively highly divergent (SHD) spaces on hyperspaces with the Vietoris topology and the Pixley-Roy topology. Moreover, we show that they are preserved under quasi-perfect countable-to-one mappings and preserved inversely under quasi-open mappings.
Access status: Abierto ,
On local semirings induced by topologies: An algebraic approach to the Collatz conjecture
(Universitat Politècnica de València, 2026-02-18) Guale, Angel; Vielma, Jorge; Escuela Superior Politécnica del Litoral
[EN] We present an algebraic approach to the Collatz conjecture by studying the topology τfon N induced by the Collatz function f , where the open sets θ ⊆ N satisfy f−1(θ) ⊆ θ.This topology, known as primal topology, turns τf into a commutative semiring. We prove that the Collatz conjecture holds if and only if τf is local. More generally, we show that any compact primal topology corresponds to a semiring that decomposes as a finite direct sum of certain local semirings and that primal compactness connectedness characterises locality. In addition, we establish that a topological space is not w-R0 if and only if its associated semiring of open sets has a unique maximal ideal such that it is an avoidance ideal of a closed set.
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On w-unicoherence, top-irreducibility and Whitney levels near the top
(Universitat Politècnica de València, 2026-02-26) Ahuatzi-Reyes, José Gerardo; Ordoñez, Norberto; Villanueva, Hugo
[EN] Given a metric continuum X, we consider C(X) as the collection of all subcontinua of X. S. López introduced the concepts of pseudo-linearity and pseudo-circularity in order to characterize continua having a positive Whitney level that is an arc or a simple closed curve. In this paper, we introduce the concepts of w-unicoherence and top-irreducibility. We study the relations between these and the well-known and more naturally-related properties defined in continuum theory, and with the concepts of pseudo-linearity and pseudo-circularity. Moreover, by using these new concepts, we obtain a new characterization of continua which have a positive Whitney level that is an arc or a simple closed curve. Also, we provide necessary conditions for a continuum to have a positive Whitney level which is a simple n-od.
Access status: Abierto ,
Collective compactness, equicompactness and the degree of nondensifiability
(Universitat Politècnica de València, 2026-02-24) García, Gonzalo
[EN] The degree of nondensifiability (DND) quantifies, in a specific sense, the distance between a bounded subset of a metric space and the Peano Continua it contains. In this paper, we use the DND to establish a quantitative version of the concepts of collective compactness and equicompactness for families of bounded linear operators between Banach spaces. Specifically, we prove inequalities that relate the DND to these notions and through some examples, we demonstrate that these inequalities are the best possible

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