Neighborhood Structures and Continuities via Cubic Sets

Abstract

Cubic sets are a very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [0, 1] and a number from [0, 1]. In this article, we first highlight some of the claims made in the previous article about cubic sets. Then, the concept of semi-coincidence in cubic sets, cubic neighborhood system according to cubic topology, and cubic bases and subbases are introduced. This article deals with a cubic closure and a cubic interior and how to obtain their various properties. In addition, cubic compact spaces and their properties are defined and a useful example is given. We mainly focus on the concept of cubic continuities and deepen our research by finding its characterization. One of the most important discoveries of this paper is determining that there is a cubic product topology induced by the projection mappings, and discovering sufficient conditions for the projection mappings to be cubic open.