Lξ-Families: Localized Topology with Applications in Edge Detection

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DOI: https://doi.org/10.28924/2291-8639-23-2025-323
Published: Nov 28, 2025
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Jamal Oudetallah, Wedad A. Alharbi, Mohammad M. Rousan, Ala Amourah, Tala Sasa, Lξ-Families: Localized Topology with Applications in Edge Detection, Int. J. Anal. Appl., 23 (2025), 323.

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Jamal Oudetallah, Wedad A. Alharbi, Mohammad M. Rousan, Ala Amourah, Tala Sasa

Abstract

We introduce and explore Lξ-families, an innovative class of localized topological structures that extends classical concepts while preserving fundamental properties. These families constitute a bridge between traditional topological objects and finer-grained local-to-global characteristics. Our construction offers a natural generalization of regular open sets through a novel localization approach that maintains critical topological invariants across various transformations and operations. This paper establishes the foundational theory of Lξ-families, proving key characterization theorems and situating them within the broader topological landscape. Our findings reveal that these structures form a complete lattice under appropriate operations and possess significant hereditary characteristics. Additionally, we demonstrate stability properties under continuous mappings and homeomorphisms, highlighting their seamless integration with established topological frameworks. Through strategically selected counterexamples, we define the boundaries of these new concepts. The theoretical architecture developed in this work creates pathways for applications in digital topology and image processing, with particularly promising implications for edge detection and boundary analysis methods. The relationships we establish between Lξ family members and classical topological concepts provide unifying perspectives across seemingly disparate notions and introduce novel tools for topological classification challenges.

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