On Nerves of Fine Coverings of Acyclic Spaces

The main results of this paper are: (1) If a space X can be embedded as a cellular subspace of \({\mathbb{R}^n}\) then X admits arbitrary fine open coverings whose nerves are homeomorphic to the n-dimensional cube Dn. (2) Every n-dimensional cell-like compactum can be embedded into (2n + 1)-dimensional Euclidean space as a cellular subset. (3) There exists a locally compact planar set which is acyclic with respect to Cech homology and whose fine coverings are all nonacyclic.