International Journal of Topology

13 pages, 303 KB

Open AccessArticle

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

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

Open AccessArticle

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 119

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.

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46 pages, 478 KB

Open AccessArticle

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 298

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 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

Open AccessArticle

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 253

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 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

Open AccessArticle

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 219

Abstract

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions

β : M \to R \times 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 : M \to R \times 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 \times Y \to Y

. 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

P_{d}^{c Θ}

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

Q_{d}^{emb} (Y, c Θ)

Θ

-constrained embeddings

β

in terms of homotopy/homology theory of spaces Y and

P_{d}^{c Θ}

. We proved also that the quasitopies of embeddings stabilize, as

d \to \infty

. Full article

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Int. J. Topol., Volume 2, Issue 3 (September 2025) – 5 articles

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