On Quasi Discrete Topological Spaces in Information Systems

This paper presents an alternative way for constructing a topological space in an information system. Rough set theory for reasoning about data in information systems is used to construct the topology. Using the concept of an indiscernibility relation in rough set theory, it is shown that the topology constructed is a quasi-discrete topology. Furthermore, the dependency of attributes is applied for defining finer topology and further characterizing the roughness property of a set. Meanwhile, the notions of base and sub-base of the topology are applied to find attributes reduction and degree of rough membership, respectively. DOI: 10.4018/jalr.2012040104 International Journal of Artificial Life Research, 3(2), 38-52, April-June 2012 39 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. ers, 1975). This has made topology one of the great unifying ideas of mathematics. Rough set theory may therefore be considered as a method for constructing a topological space using indiscernibility relation on the universe. In a reverse process, we can generalize the notions of rough sets based on the topological space (Herawan & Deris, 2009). Definable sets, i.e. a union of one or more equivalence classes are substituted by open sets in defining the lower approximations, and by closed sets in defining upper approximations (Yao, 1998, 2001). Lashin et al. (2005) presented rough set theory in general topological space and described rough set theory in the topology of binary relation by defining (right) neighborhood and investigate the knowledge representations (granular structures) and processing of binary relations in the style of rough set theory. Zhu (2007) discussed covering-based rough sets from the topological view, defined a new type of covering-based rough sets from a topological concept called neighborhood and established axiomatic systems for the lower and the upper approximations operations. Allam et al. (2008) proposed and discussed some methods for generating topologies are using binary relations. They also investigated some properties of these topologies and obtained a quasi-discrete topology from a symmetric relation instead of an equivalence relation. Zhang et al. (2010) investigated the role of topological De Morgan algebra in the theory of rough sets. However, in Lashin et al. (2005), Zhu et al. (2007), Allam et al. (2008), and Zhang et al. (2010), the concept of topological rough set in an information system is not discussed. Salama (2010) presented a new method of data decomposition to avoid the necessity of reasoning from data with missing attribute values. A general binary relation is defined on the original incomplete dataset. This binary relation generates data subsets without missing values. These data subsets are used to generate a topological base relation which decomposes datasets. Instead of missing values in information system, in this paper, we apply “standard” rough set theory for topological space of reasoning from complete dataset (without missing attribute values). We begin with the using of rough set theory for constructing a topological space in information systems. The results show that: the family of all definable sets is a quasi-discrete topology on the universe, the partition induced by indiscernibility relation in information systems is the base for the related, and the union of partitions induced by singleton attribute is a sub-base for the related base. Further, we use the dependency of attributes in information systems for determining the finer topology and this can be used to characterize the roughness property of a set. Finally, we show that the attributes reduction and the degree of rough membership can be defined involving the sub-base and base for the related topology, respectively. Although some results are presented, a major part of this paper is devoted to reveal interconnections between rough set theory and topological spaces in information systems. This paper is organized as follows. First, we describe the fundamental concepts of rough set theory for reasoning about data. Then a description of some notions in topological spaces. Afterwards we describe the main results, rough set theory for topological spaces in information systems. Finally, we conclude our work in the last section.