Čech systems that induce approximate inverse systems

Abstract

In previous work we showed that if a paracompact Hausdorff space contains a nontrivial component, then none of the Čech 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. Mardešić, 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 Čech 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 Čech 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.