Journal of Advances in Mathematics and Computer Science

This study develops a mapping framework for pentapartitioned neutrosophic topological spaces by introducing and systematically examining pentapartitioned neutrosophic \(\beta\)-open and \(\beta\)-closed mappings. Fundamental properties and characterisations of these mappings are established, and their relationships with the corresponding generalised open and closed mappings are investigated. The study also introduces pentapartitioned neutrosophic \(\beta\)-homeomorphisms and pentapartitioned neutrosophic \(\beta\)-completely homeomorphisms as generalised forms of topological equivalence within the pentapartitioned neutrosophic setting. Preservation, composition, and characterisation results are derived for the proposed mappings and homeomorphisms. In particular, the findings describe the behaviour of pentapartitioned neutrosophic topological spaces under \(\beta\)-open and \(\beta\)-closed mappings and identify conditions under which the associated homeomorphism concepts are retained. The results organise several related mapping notions within a common theoretical framework and clarify their implications for structures that represent truth, contradiction, ignorance, unknown information, and falsity. By extending existing work on pentapartitioned neutrosophic topology, the study provides a basis for further investigation of generalised continuity, mapping theory, and topological equivalence under indeterminate and inconsistent information. The proposed framework also records implication relations among the mapping classes and supplies examples that distinguish converses that do not hold in general. The contribution is primarily theoretical and is confined to the definitions, properties, examples, and relationships developed in the manuscript.