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].

This work was supported by Tokyo Denki University Science Promotion Fund (Q18K-01).