Algebraic Operations and Topological Framework on Order (k) and Degree (j) Sets

Abstract

The operations such as union, intersection, complement are extended to proposed (k, j)-picture sets. These operations play an important role in developing the theoretical framework of    (k, j)-picture set theory. The fundamental properties that exist in classical set theory are also examined within this new setting. By studying these properties, similarities and differences between classical sets and (k, j)-picture sets can be identified. The behaviour of these operations helps in understanding the structure of (k, j)-picture sets more clearly. In particular, they lead to introduce the topological structure on a family of (k, j)-picture sets. This topology is defined over a non-empty universe X and helps to study continuity and related concepts in the context of (k, j)-picture set theory.