On nerves of fine coverings of acyclic spaces (Q2018703)

We can think of a cube \(D^n\subset \mathbb R^n\) and \(\mathcal{U}\) an open covering. To each \(\mathcal{U}\) is assigned the nerve \(N(\mathcal{U})\) and if \(\mathcal{U}\) is sufficiently fine we may ask what property the nerve has, like is it homeomorphic to \(D^n\)? The main result generalizes this thinking. It considers a space \(X\) that can be embedded in \(\mathbb R^n\) as a cellular subspace (i.e. as a nested sequence of \(n\)-cubes) and proves that it admits arbitrarily fine open coverings whose nerves are homeomorphic to the \(n\)-dimensional cube \(D^n\). So this happens with fine coverings to of \(D^n\subset \mathbb R^n\) too. From the above statement we see that one condition is that the considered space \(X\) is a subset of the Euclidean space. An old result says that every \(n\)-dimensional compact metrizable space can be embedded into \(\mathbb R^{2n+1}\). As the next result the authors prove that every \(n\)-dimensional cell-like compact space embeds into \(\mathbb R^{2n+1}\) as a cellular subset. Consequently, \(n\)-dimensional cell-like spaces admits arbitrarily fine coverings whose nerves are homeomorphic to \(D^{2n+1}\). In addition two examples are given of locally compact planar sets acyclic with respect to Čech homology all of whose fine coverings are non-acyclic.