用多余的一个覆盖：看到拓扑

摘要：

我们已经开始了对网格域上覆盖空间拓扑的研究。这个空间有以下的限制：虽然所有的覆盖代理可以在域上自由移动（我们允许重叠），但它们的并集必须覆盖整个域。成功覆盖域需要最小数量的覆盖代理$N$。在本文中，我们证明了这个空间在二维网格域上具有$N+1$覆盖的美丽拓扑结构，该空间的拓扑具有$1$维复形的同伦类型，而与域的形状无关.给出了连接空间拓扑与区域拓扑的欧拉示性数公式。

英文摘要：

We have initiated the study of topology of the space of coverings on grid domains. The space has the following constraint: while all the covering agents can move freely (we allow overlapping) on the domain, their union must cover the whole domain. A minimal number $N$ of the covering agents is required for a successful covering of the domain. In this paper, we demonstrate beautiful topological structures of this space on grid domains in 2D with $N+1$ coverings, the topology of the space has the homotopy type of $1$ dimensional complex, regardless of the domain shape. We also present the Euler characteristic formula which connects the topology of the space with that of the domain itself.

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