Abstract 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 τ R from a relation R is exactly the transitive closure of I ∪ R , where I is the relation of equality, and that the topology induced by the relation R τ from a topology τ is exactly τ if and only if τ is quasi-discrete. Then we examine special topological properties of τ 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.

中文翻译：

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拓扑与广义粗糙集的关系

摘要 拓扑可用作数据库中知识发现的工具，并在粗糙集理论中得到了深入研究。拓扑和二元关系可以相互归纳。在本文中，我们主要研究由任意二元关系引起的拓扑。我们首先考虑这种拓扑的基本性质，证明由关系 R 由拓扑 τ R 导出的关系正是 I ∪ R 的传递闭包，其中 I 是等式关系，并且由关系导出的拓扑当且仅当 τ 是准离散的，拓扑 τ 中的 R τ 恰好是 τ。然后我们检查由二元关系 R 引起的 τ R 的特殊拓扑性质。特别是，我们获得了分离公理的等效描述，并提供了拓扑性质的表征，例如紧致性、