\(\alpha_\sigma\)-Sets: A Novel Framework for Generalized Topological Structures with Applications in Digital Topology

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DOI: https://doi.org/10.28924/2291-8639-23-2025-271
Published: Nov 5, 2025
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Jamal Oudetallah, Rehab Alharbi, Ala Amourah, Iqbal M. Batiha, Tala Sasa, \(\alpha_\sigma\)-Sets: A Novel Framework for Generalized Topological Structures with Applications in Digital Topology, Int. J. Anal. Appl., 23 (2025), 271.

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Jamal Oudetallah, Rehab Alharbi, Ala Amourah, Iqbal M. Batiha, Tala Sasa

Abstract

This paper introduces ασ-sets as a new class of generalized topological structures that extend α-open sets to countable unions with controlled boundary behavior. We prove these structures form a σ-algebra under union operations and exhibit strong hereditary properties in subspaces. Our investigation establishes fundamental preservation properties under continuous mappings and homeomorphisms, with characteristic behavior in product spaces. The framework bridges classical topological concepts with refined local-to-global properties while preserving critical topological invariants. We demonstrate applications in digital topology and image processing, particularly for texture analysis and pattern recognition, where ασ-sets effectively capture complex boundary behaviors. Through counterexamples and characterization theorems, we precisely position these structures within the broader topological landscape, providing new tools for topological classification problems in image analysis.

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