Topological characterizations to three types of covering approximation operators

TL;DR

This paper explores the topological properties of three types of covering approximation operators in rough set theory, establishing conditions under which they induce specific topologies and comparing these spaces.

Contribution

It provides new topological characterizations of covering approximation operators and links their properties to induced topologies, advancing the theoretical understanding of rough sets.

Findings

Topologies induced by certain covering approximation operators are equivalent.

Conditions are identified for approximation operators to be interior or closure operators.

Comparison of the topological spaces induced by different covering approximation operators.

Abstract

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Topology, one of the most important subjects in mathematics, provides mathematical tools and interesting topics in studying information systems and rough sets. In this paper, we present the topological characterizations to three types of covering approximation operators. First, we study the properties of topology induced by the sixth type of covering lower approximation operator. Second, some topological characterizations to the covering lower approximation operator to be an interior operator are established. We find that the topologies induced by this operator and by the sixth type of covering lower approximation operator are the same. Third, we study the conditions which make the first type of covering upper approximation operator be a closure operator, and find that the topology induced by the operator is the same as the topology induced by the fifth type of covering upper approximation operator. Forth, the conditions of the second type of covering upper approximation operator to be a closure operator and the properties of topology induced by it are established. Finally, these three topologies space are compared. In a word, topology provides a useful method to study the covering-based rough sets.