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Top1. Introduction
Equivalence relations (ERs) are used to define Rough sets (RS) and are modeled to capture uncertainty in data (Pawlak, 1982). Successful application of rough sets depends upon two notions; the topological characterization and the accuracy measure as observed by Pawlak (1991). Granules are the smallest addressable units of data. The study of granules is beneficial from the point of view that the characteristics of all the elements in a granule are similar in view of the real life applications. It makes the study simpler and concise. Zadeh in 1979 introduced Granular computing (GC) in the context of fuzzy sets and again it was revived by him in 1997. Study of granules reduces the complexity of algorithms without affecting the goals of study. The first definition of rough sets uses unigranular structure from the GC point of view. However, the notion was generalised by introducing rough sets on a class of ERs. This is perhaps the first instance of mutigranularities. In many real-life applications, it is highly required to handle multiple granularity of a universe of discourse taken simultaneously. In 2006 the first type of mutigranularities, called the optimistic multigranulation based upon rough sets was introduced in and later in the second type called pessimistic multigranulation based upon rough sets was introduced and studied (Qian & Liang, 2006; Qian et al., 2007; Qian et al., 2010). Since then, multigranular rough sets for other rough set models like fuzzy rough sets, rough sets on incomplete approximation spaces, covering based rough sets (CBRS) and fuzzy approximation spaces (FAS) have been studied by several authors. Several types of CBRS have been introduced in the literature. Topological approach to CBRS is introduced and studied in (Tripathy & Nagaraju, 2012; Tripathy & Nagaraju, 2015; Qian et al., 2007; De, 1999; De et al., 2003; Zhu, 2002; Zhu & Wang, 2007; Zhu, 2007).
The latest among the extensions of the multigranulation concept is to rough set model is to the notion of rough sets on FAS (Tripathy & Bhambani, 2018). A formal definition is to be provided in the next section. In this case a fuzzy proximity relation is taken as the basis for defining a rough set instead of an ER. The rough set models are associated with two types of properties; the algebraic properties and the topological properties. The whole set comes under consideration with respect to topological properties, whereas the individual elements come under the algebraic properties. Two important concepts associated with the study of rough sets are; their accuracy measure and categories (termed as categories by Pawlak). Both the properties should be considered simultaneously for dealing with any application successfully. Depending upon the structure of the lower and upper approximations of a set it can be categorized under 4 categories. Here we focus to deal with MGRS on FAS. It is interesting to find the categories of set theoretic operations of union, intersection and complement of a set. It has enough applications to the classification of sets as has been shown in Tripathy (2006, 2010). So, this study will pave the way for some analysis in the direction of these two papers. Classifications are the building blocks of rough sets in its basic form and the studies in confirm the statement of Pawlak that the complement in bi-valued logic is different from the notion of complements in case of multi-valued logic cab be highlighted further (Tripathy et al., 2006; Tripathy & Mitra, 2010).
The arrangement of the different sections of this paper henceforth is that the second section presents various concepts to be used in the paper. We introduce rough sets on FAS section 3 along with their properties. Topological properties in the context of basic rough sets are discussed in section 4. Next, we deal with the topological properties of MGRS on FAS.by determining the categories of sets obtained by the three set theoretic operations of union, intersection and complement in section 5. This is followed by concluding remarks on the work done in the next section.