On Topologies Defined by Binary Relations in Rough Sets

Abstract

We consider relationship between binary relations in approximation spaces and topologies defined by them. In any approximation space (X, R), a reflexive closure$$R:{\omega }$$ determines an Alexandrov topology$$\mathcal {T}_{(R_{\omega })}$$ and, for any Alexandrov topology$$\mathcal {T}$$ on X, there exists a reflexive relation$$R:{\mathcal {T}}$$ such that$$\mathcal {T}= \mathcal {T}_R$$. From the result, we also obtain that any Alexandrov topology satisfying (clop), A is open if and only if A is closed, can be characterized by reflexive and symmetric relation. Moreover, we provide a negative answer to the problem left open in [1].