AI 中文总结
本文探讨能否用粗糙族刻画$T_2$拓扑空间，证明拓扑空间是$T_2$当且仅当非粗糙拓扑空间。通过粗糙族定义粗糙内部等概念，产生粗糙拓扑与同胚，构造例子展示新颖性，还扩展了紧致性和连通性概念。
AI 中文摘要
最近，在[莱奥内蒂，P.，《凸分析杂志》32(4):1083 - 1090，2025]中引入了粗糙族的概念，以探索拓扑空间中的粗糙理想收敛，其中序列的极限可能不唯一。这就引发了是否能用粗糙族来刻画$T_2$拓扑空间的问题。在本文中，我们证明一个拓扑空间是$T_2$当且仅当它永远不可能是粗糙拓扑空间。在此背景下，我们首先从粗糙族的角度引入集合的粗糙内部和粗糙闭包的概念，进而定义粗糙开集（粗糙闭集）。由此产生了一种新的拓扑，称为粗糙拓扑，以及粗糙同胚。我们的主要贡献在于展示了这类新拓扑的新颖之处；特别是，我们明确构造了几个例子，确保两个非同胚空间在一定粗糙度下可以是粗糙同胚的。此外，我们扩展了紧致性和连通性的概念，我们在这些领域的发现与现有文献不同，简而言之，提供了新的见解和观点。
英文摘要
Very recently, the notion of rough family has been introduced in [Leonetti, P., J. Convex Anal. 32(4):1083-1090, 2025] to explore rough ideal convergence in topological spaces where the limit of a sequence may not be unique. This raises the question of whether $T_2$ topological spaces can be characterized using rough families. In this article, we prove that a topological space is $T_2$ if and only if it can never be a rough topological space. In this context, we first introduce the notions of rough interior and rough closure of a set from the perspective of a rough family, which leads to the definition of rough open sets (rough closed sets). As a consequence, we generate a new topology, termed rough topology, as well as rough homeomorphism. Our main contribution presents the novelty of this new class; in particular, we explicitly construct several examples which ensure that two non-homeomorphic spaces can be roughly homeomorphic under certain roughness. Additionally, we extend the concepts of compactness as well as connectedness, where our findings diverge from existing literature in these areas, in a nutshell, providing new insights and perspectives.
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