Quite recently, Wu and Liu (2020) [11] raised an open problem when they discussed the relationships between generalized rough sets and topologies. Said precisely, each binary relation generates a topology through the lower rough approximation operator, then for two binary relations on the same set, is there a sufficient and necessary condition such that the union of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. In this note, we give a positive answer to this problem and provide a union-join condition. Furthermore, note that the main reason that the expected identity hold not consists in the fact that the union of two topologies is not a topology, in general. Then replacing the union of two topologies with the supremum of them, we give a sufficient and necessary condition (supremum-join condition) such that the supremum of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. We verify that the supremum-join condition is much simpler than the union-join condition. At last, we give a characterization on discrete topology generated by a binary relation using the sober separation.

最近，Wu和Liu（2020）[11]在讨论广义粗糙集和拓扑之间的关系时提出了一个开放的问题。准确地说，每个二元关系通过较低的粗略近似算子生成一个拓扑，然后对于同一集合上的两个二元关系，存在充分必要的条件，使得通过两个二元关系生成的拓扑的并集与生成的拓扑相同拓扑由两个二进制关系的联接组成。在本说明中，我们对这个问题给出了肯定的答案，并提供了一个联合联接条件。此外，请注意，预期身份不成立的主要原因在于，通常两个拓扑的结合不是拓扑。然后用它们的最高取代两个拓扑的并集，我们给出一个充分必要的条件（最高联接条件），以使通过两个二元关系生成的拓扑的最高值与通过两个二元关系的连接生成的拓扑的相同。我们验证了最高联接条件比联合联接条件简单得多。最后，我们给出了使用清醒分离由二元关系生成的离散拓扑的特征。