Gauges, pregauges and completions: some theoretical aspects of near and rough set approaches to data

• Authors: Marcin Wolski
• Venue: RSKT'11 Proceedings of the 6th international conference on Rough sets and knowledge technology
• Year: 2011

The theory of near sets and the theory of rough sets share a common metric root. Since a probe function and an equivalence relation can be regarded as a pseudometric on U, in actual fact the underlying structure of both theories is a family of pseudometrics. The same starting point one can find in metric topology (e.g. [2]): an arbitrary family of pseudometrics is called a pregauge structure and when this family additionally separates all points, it is called a gauge structure. Pregauge structures characterise all completely regular spaces, whereas gauge structures correspond to all Hausdorff completely regular spaces (often called gauge spaces). In consequence, a perceptual system and an information system can be regarded as both pregauge structures and as topological completely regular spaces. In the paper, we would like to make a step towards gauge structures. A perceptual system or an approximation space does usually not separate all points, therefore we introduce the concept of a completion of a pregauge, but on the relational level: that is, a pregauge is regarded as a family E of equivalence relations, and a completion of ε is a relation R which added to ε, makes this family separate all points of U. If R is an equivalence relation, then ε ∪ R can be converted into a gauge structure and the corresponding topology will be Hausdorff completely regular. Since, in data analysis, U is finite, this topology will be discrete. Therefore, our aim is actually to find weaker topologies than gauge spaces. To this end, we allow R to be a tolerance relation and we consider topologies on the pregauge structure and on its completion, separately. At the end we present a simple application of these topologies.