Abstract In previous work we showed that if a paracompact Hausdorff space contains a nontrivial component, then none of the Cech systems of the nerves of its open covers can be an approximate (inverse) system. This extended a theorem of the first author who, in answering a question of S. Mardesic, proved this result in the restricted case that the given Hausdorff space was arc-like (and hence is nontrivial, compact and connected). We will demonstrate that the same is true for a nonempty discrete space: none of the Cech systems of the nerves of its open covers can be an approximate (inverse) system. In our main theorem, we are going to show in contradistinction that the completely opposite phenomenon occurs in the case that the space in question is strongly 0-dimensional (meaning that it has covering dimension 0) and has a limit point by proving that at least one of its Cech systems can support the structure of an approximate system. We will also show that such a space can be written as the inverse limit, in the classical sense, of an inverse system of discrete spaces.

中文翻译：

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Čech 系统诱导近似逆系统

摘要 在之前的工作中，我们表明，如果一个超紧致的 Hausdorff 空间包含一个非平凡的分量，那么它的开盖神经的 Cech 系统都不能是近似（逆）系统。这扩展了第一作者的定理，他在回答 S. Mardesic 的问题时，在给定的 Hausdorff 空间是弧状（因此是非平凡的、紧凑的和连通的）的受限情况下证明了这个结果。我们将证明对于非空离散空间也是如此：其开盖神经的 Cech 系统都不能是近似（逆）系统。在我们的主定理中，我们将相反地证明，在所讨论的空间是强 0 维的（意味着它覆盖 0 维）并且有一个极限点的情况下，会发生完全相反的现象，通过证明它的至少一个 Cech 系统可以支持近似系统的结构。我们还将证明，这样的空间可以写成经典意义上的离散空间逆系统的逆极限。