α_σ-Sets: A Novel Framework for Generalized Topological Structures with Applications in Digital Topology

This paper introduces α_σ-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 preser-vation 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 α_σ-sets effectively capture complex boundary behaviors. Through counterex-amples and characterization theorems, we precisely position these structures within the broader topological landscape, providing new tools for topological classification problems in image analysis.