Ideal-convergence classes

Georgiou, D.N.; Iliadis, S.D.; Megaritis, A.C.; Prinos, G.A.

doi:10.1016/j.topol.2017.02.045

Cited by 10 publications

(4 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Specifically, we examine the necessary and sufficient conditions that a fuzzy function ideal convergence class L, on a non-empty set X, should fulfill to determine a unique fuzzy topology δ on X such that I-convergence (L) coincides with I-convergence, with respect to δ. All the results obtained here are parallel to and extend those given in [18,19,21], for the ordinary topology, while simultaneously simplify the exposition and the underlying theory. In order to increase the utility of the present work, future research options may include the extension of the lattice background, L, to completely distributive lattices with an ordering-reversing involution as a tool to investigate the notion of L-fuzzy function ideal convergence and its applications in the more general context of L-fuzzy topological spaces.…”

Section: Discussionsupporting

confidence: 74%

“…Lahiri and P. Das, in [16,17], investigated the notion of the ideal convergence of sequences and nets in topological spaces. In the context of net ideal convergence, the authors of [18,19] provided a modified version of J. Kelley's classical theorem [20] (p. 74, Theorem 9) for convergence classes. More precisely, in [18] they considered a non-empty set, X, and a class, C, consisting of triples of the form ((s d ) d∈D , x, I), where (s d ) d∈D is a net with a domain on the directed set, D, and values on X, I is a D-admissible ideal on D and x ∈ X and provided a set of axioms on the class, C, which are necessary and sufficient to ensure the existence of a unique topology, τ, on X, such that ((s d ) d∈D , x, I) ∈ C iff (s d ) d∈D I-converges to x, relative to the topology, τ.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Notion of Fuzzy Function Ideal Convergence

Georgiou,

Prinos

2024

Mathematics

View full text Add to dashboard Cite

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.

show abstract

Section: Discussionsupporting

confidence: 74%

Section: Introductionmentioning

confidence: 99%

A New Notion of Fuzzy Function Ideal Convergence

Georgiou,

Prinos

2024

Mathematics

View full text Add to dashboard Cite

show abstract

“…Later B.K. Lahiri and P. Das in [12] discussed convergence in I and in I * and investigate some additional results related to mentioned concepts [4,[8][9][10][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

\(\mathcal{I}^{\mathcal{K}}\)-SEQUENTIAL TOPOLOGY

Behmanush,

Küçükaslan

2023

UMJ

View full text Add to dashboard Cite

In the literature, \(\mathcal{I}\)-convergence (or convergence in \(\mathcal{I}\)) was first introduced in [11]. Later related notions of \(\mathcal{I}\)-sequential topological space and \(\mathcal{I}^*\)-sequential topological space were introduced and studied. From the definitions it is clear that \(\mathcal{I}^*\)-sequential topological space is larger(finer) than \(\mathcal{I}\)-sequential topological space. This rises a question: is there any topology (different from discrete topology) on the topological space \(\mathcal{X}\) which is finer than \(\mathcal{I}^*\)-topological space? In this paper, we tried to find the answer to the question. We define \(\mathcal{I}^{\mathcal{K}}\)-sequential topology for any ideals \(\mathcal{I}\), \(\mathcal{K}\) and study main properties of it. First of all, some fundamental results about \(\mathcal{I}^{\mathcal{K}}\)-convergence of a sequence in a topological space \((\mathcal{X} ,\mathcal{T})\) are derived. After that, \(\mathcal{I}^{\mathcal{K}}\)-continuity and the subspace of the \(\mathcal{I}^{\mathcal{K}}\)-sequential topological space are investigated.

show abstract

“…Salát et al [34] extended the notion of summability fields of an infinite matrix of operators A with the help of the notion of I-convergence, that is, the notion of I-summability and introduced new sequence spaces c J A and m J A , the I-convergence field and bounded I-convergence field of an infinite matrix A, respectively. For further details on ideal convergence, we refer to [1,19,24], etc.…”

Section: Introductionmentioning

confidence: 99%