Separation Axioms, Connectedness and Metrizability in Neutrosophic Topological Spaces

Abstract

 This paper investigates separation axioms, connectedness and metrizability in neutrosophic topological spaces. Neutrosophic topology extends classical and fuzzy topology by assigning to each element a triple of truth-membership, indeterminacy and falsity-membership values. The paper first reviews the basic structure of neutrosophic sets, neutrosophic points, neutrosophic open sets and neutrosophic closed sets. It then develops a systematic account of neutrosophic separation axioms, including -, -, -, regularity and normality. The implications among these separation axioms are established, and examples are supplied to show that several converses fail. The connectedness section characterizes neutrosophic connectedness through the absence of proper neutrosophic clopen sets and relates it to continuous images and subspaces. The compactness section records basic covering properties and their interaction with continuous maps. The metrizability section introduces neutrosophic metrics, neutrosophic metric balls and sufficient conditions under which a neutrosophic topology is induced by a neutrosophic metric. Comparative tables and formal diagrams are included to distinguish classical, fuzzy, intuitionistic fuzzy and neutrosophic topological structures. The resulting article provides a consolidated theorem-proof framework suitable for further research in neutrosophic topology and related generalized topological systems.