On continuity and categorical property of interval-valued topological spaces

An interval set (or an interval-valued set), is a special set, which is an effective tool for illustrating and describing obscure information systems and partially known problems. Recently, Kim et al.[5] defined the topological structure for interval-value sets and studied many properties of them. In this work, we discuss some characteristics and relations of continuity in interval-valued topological spaces with some necessary illustrative examples. Then we provide a categorical framework for interval-valued topological spaces IV-T OP. Many functors and subcategories of IV-T OP are defined and studied. Furthermore, the relationships between the IV-T OP and its subcategories are investigated. We show that the category T OP is isomorphic to the category IV-T OP1. Moreover, we demonstrate that T OP and IV-T OP1 are bireflective full subcategories of IV-T OP.