Abstract The present paper focuses on the studies of the topological (interior and closure) operators of the M-rough set structure ( M ⁎ , M ⁎ ) and the Marcus-Wyse (MW-, for brevity) topological rough set structure ( D M − , D M + ) . While the concept approximations ( M ⁎ , M ⁎ ) are topological operators, it is well known that the function D M − is not an interior operator and D M + is not a closure one either. Hence an issue can be posed as follows: What condition makes D M − (resp. D M + ) an interior (resp. a closure) operator? This paper proposes a new frame addressing the issue, the so-called ( Δ M − , Δ M + ) instead. This new version smoothly matches with ( M ⁎ , M ⁎ ) w.r.t. the topological operators. Furthermore, both ( M ⁎ , M ⁎ ) and ( Δ M − , Δ M + ) are shown to have many theoretical and mathematical properties in common. Therefore, they can efficiently be used in applied sciences with a strong combination. Besides, they can support certain decision rules without any limitations of studying something continuous or discrete (or digital) from the viewpoint of covering rough set theory. In this paper each of a universe U and a target set X of U need not be finite, and a covering C is locally finite.

中文翻译：

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MW-拓扑粗近似的拓扑算子

摘要 本文重点研究 M-粗糙集结构 (M ⁎ , M ⁎ ) 和 Marcus-Wyse (MW-, 为简洁起见) 拓扑粗糙集结构 ( DM - , DM + )。虽然概念近似（ M ⁎ , M ⁎ ）是拓扑算子，但众所周知，函数 DM - 不是内部算子，DM + 也不是闭包算子。因此，可以提出如下问题：什么条件使 DM − (resp. DM + ) 成为内部（resp. aclosure）算子？本文提出了一个新的框架来解决这个问题，即所谓的 (Δ M − , Δ M + )。这个新版本与 ( M ⁎ , M ⁎ ) 与拓扑算子平滑匹配。此外，( M ⁎ , M ⁎ ) 和 ( Δ M - , Δ M + ) 都被证明具有许多共同的理论和数学特性。所以，它们可以通过强大的组合有效地用于应用科学。此外，它们可以支持某些决策规则，而不受从覆盖粗糙集理论的角度研究连续或离散（或数字）事物的任何限制。在本文中，宇宙 U 和 U 的目标集 X 中的每一个都不必是有限的，覆盖 C 是局部有限的。