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Int. J. Topol., Volume 2, Issue 3 (September 2025) – 7 articles

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61 pages, 571 KB  
Article
Topological Types of Convergence for Nets of Multifunctions
by Marian Przemski
Int. J. Topol. 2025, 2(3), 15; https://doi.org/10.3390/ijt2030015 - 11 Sep 2025
Viewed by 295
Abstract
This article proposes a unified concept of topological types of convergence for nets of multifunctions between topological spaces. Any kind of convergence is representable by a (2n + 2)-tuple, n = 0, 1, …, of two special functions u and l, such [...] Read more.
This article proposes a unified concept of topological types of convergence for nets of multifunctions between topological spaces. Any kind of convergence is representable by a (2n + 2)-tuple, n = 0, 1, …, of two special functions u and l, such that their compositions ul and lu create the Choquet supremum and infimum operations, respectively, on the filters considered in terms of the upper Vietoris topology on the range hyperspace of the considered multifunctions. Convergence operators are defined by establishing the order of composition of the functions from such (2n + 2) tuples. An allocation of places for the two distinguished functions in a convergence operator reflects the structure of the used (2n + 2)-tuple. A monoid of special three-parameter functions called products describes the set of all possible structures. The monoid of products is the domain space of the convergence operators. The family of all convergence operators forms a finite monoid whose neutral element determines the pointwise convergence and possesses the structure determined by the neutral element of the monoid of products. We demonstrate the construction process of every convergence operator and show that the notions of the presented concept can characterize many well-known classical types of convergence. Of particular importance are the types of convergence derived from the concept of continuous convergence. We establish some general theorems about the necessary and sufficient conditions for the continuity of the limit multifunctions without any assumptions about the type of continuity of the members of the nets. Full article
19 pages, 2667 KB  
Article
Theoretical Aspects of Topology and Successful Applications to Glasses and Proteins
by Punit Boolchand, James Charles Phillips, Matthieu Micoulaut and Aaron Welton
Int. J. Topol. 2025, 2(3), 14; https://doi.org/10.3390/ijt2030014 - 9 Sep 2025
Viewed by 613
Abstract
The origin of glass formation has been one of the greatest mysteries of science. The first clues emerged in GexSe1-x glasses, where the bond-stretching and bond angle-bending constraints are countable, and it was found that the most favorable compositions for [...] Read more.
The origin of glass formation has been one of the greatest mysteries of science. The first clues emerged in GexSe1-x glasses, where the bond-stretching and bond angle-bending constraints are countable, and it was found that the most favorable compositions for glass formation involved matching constraints with the degrees of freedom. Modulated-Differential Scanning Calorimetric (MDSC) studies on GexSe1-x chalcogenide glasses revealed two elastic phase transitions—a stiffness transition at x = 0.20 and a stress transition at x = 0.26—leading to the observation of three topological phases: a flexible phase at x < 0.20, an intermediate phase in the 0.20 < x < 0.26 range, and a stressed–rigid phase for compositions x > 0.26. The three topological phases (TPs) have now been generically observed in more than two dozen chalcogenides and modified oxide glasses. In proteins, the transition from the unfolded (flexible) to the folded (isostatically rigid intermediate) phase represents the stiffness transition. Self-organization causes proteins to display a dynamic reversibility of the folding process. The evolutions of protein dynamics may also exhibit stiffness phase transitions similar to those seen in glasses. Full article
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13 pages, 303 KB  
Article
On Connected Subsets of a Convergence Space
by Bryan Gerardo Castro Herrejón and Frédéric Mynard
Int. J. Topol. 2025, 2(3), 13; https://doi.org/10.3390/ijt2030013 - 27 Aug 2025
Viewed by 1023
Abstract
Though a convergence space is connected if and only if its topological modification is connected, connected subsets of a convergence space differ from those of its topological modification. We explore which subsets exhibit connectedness for the convergence or for the topological modification. In [...] Read more.
Though a convergence space is connected if and only if its topological modification is connected, connected subsets of a convergence space differ from those of its topological modification. We explore which subsets exhibit connectedness for the convergence or for the topological modification. In particular, we show that connectedness of a subset is equivalent for a convergence or for its reciprocal modification and that the largest set enclosing a given connected subset of a convergence space is the adherence of the connected set for the reciprocal modification of the convergence. Full article
12 pages, 256 KB  
Article
Bornological Approach Nearness
by Dieter Leseberg and Zohreh Vaziry
Int. J. Topol. 2025, 2(3), 12; https://doi.org/10.3390/ijt2030012 - 7 Aug 2025
Viewed by 257
Abstract
We introduce the notion of bornological approach nearness as a unified extension of various classical nearness structures. By redefining completeness within this framework, we establish a generalized version of the Niemytzki–Tychonoff theorem. Our results not only extend known compactness criteria in nearness spaces [...] Read more.
We introduce the notion of bornological approach nearness as a unified extension of various classical nearness structures. By redefining completeness within this framework, we establish a generalized version of the Niemytzki–Tychonoff theorem. Our results not only extend known compactness criteria in nearness spaces but also offer a new perspective that incorporates boundedness and bornological methods in the theory of approach spaces. Full article
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46 pages, 478 KB  
Article
Extensions of Multidirected Graphs: Fuzzy, Neutrosophic, Plithogenic, Rough, Soft, Hypergraph, and Superhypergraph Variants
by Takaaki Fujita
Int. J. Topol. 2025, 2(3), 11; https://doi.org/10.3390/ijt2030011 - 21 Jul 2025
Viewed by 549
Abstract
Graph theory models relationships by representing entities as vertices and their interactionsas edges. To handle directionality and multiple head–tail assignments, various extensions—directed, bidirected, and multidirected graphs—have been introduced, with the multidirected graph unifying the first two. In this work, we further enrich this [...] Read more.
Graph theory models relationships by representing entities as vertices and their interactionsas edges. To handle directionality and multiple head–tail assignments, various extensions—directed, bidirected, and multidirected graphs—have been introduced, with the multidirected graph unifying the first two. In this work, we further enrich this landscape by proposing the Multidirected hypergraph, which merges the flexibility of hypergraphs and superhypergraphs to describe higher-order and hierarchical connections. Building on this, we introduce five uncertainty-aware Multidirected frameworks—fuzzy, neutrosophic, plithogenic, rough, and soft multidirected graphs—by embedding classical uncertainty models into the Multidirected setting. We outline their formal definitions, examine key structural properties, and illustrate each with examples, thereby laying groundwork for future advances in uncertain graph analysis and decision-making. Full article
28 pages, 338 KB  
Article
Superhypermagma, Lie Superhypergroup, Quotient Superhypergroups, and Reduced Superhypergroups
by Takaaki Fujita
Int. J. Topol. 2025, 2(3), 10; https://doi.org/10.3390/ijt2030010 - 8 Jul 2025
Viewed by 388
Abstract
Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of [...] Read more.
Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of elements, giving rise to hypermagmas, hypergroups, and Lie hypergroups, along with their variants such as quotient, reduced, and fuzzy hypergroups. In this work, we introduce the concept of superhyperstructures, obtained by iterating the powerset construction, and develop the theory of superhypermagmas and Lie superhypergroups. We further define and analyze quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, exploring their algebraic properties and interrelationships. Full article
(This article belongs to the Special Issue Feature Papers in Topology and Its Applications)
48 pages, 944 KB  
Article
Spaces of Polynomials as Grassmanians for Immersions and Embeddings
by Gabriel Katz
Int. J. Topol. 2025, 2(3), 9; https://doi.org/10.3390/ijt2030009 - 24 Jun 2025
Viewed by 367
Abstract
Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions β:MR×Y of compact n-manifolds M such that β(M) avoids some priory chosen closed poset Θ of tangent patterns to [...] Read more.
Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions β:MR×Y of compact n-manifolds M such that β(M) avoids some priory chosen closed poset Θ of tangent patterns to the fibers of the obvious projection π:R×YY. Then, for a fixed Y, we introduced an equivalence relation between such β’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces PdcΘ of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset Θ, play the classical role of Grassmanians. We computed the quasitopy classes Qdemb(Y,cΘ) of Θ-constrained embeddings β in terms of homotopy/homology theory of spaces Y and PdcΘ. We proved also that the quasitopies of embeddings stabilize, as d. Full article
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