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The relationships between topologies and generalized rough sets. (English) Zbl 1434.68578

Summary: Topology can be used as a tool for knowledge discovery in databases and has been studied intensively in rough set theory. Topology and binary relations can be induced from each other. In this paper, we mainly study topologies induced by arbitrary binary relations. We first consider the elementary properties of such topologies, show that the relation induced by the topology \(\tau_R\) from a relation \(R\) is exactly the transitive closure of \(I \cup R\), where \(I\) is the relation of equality, and that the topology induced by the relation \(R_\tau\) from a topology \(\tau\) is exactly \(\tau\) if and only if \(\tau\) is quasi-discrete. Then we examine special topological properties of \(\tau_R\) induced by a binary relation \(R\). In particular, we obtain equivalent descriptions of separation axioms and provide characterizations of topological properties such as compactness, second countability and connectedness.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
54A99 Generalities in topology
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