  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
$\mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded
into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There
exists a locally compact planar set which is acyclic with respect to \v{C}ech
homology and whose fine coverings are all nonacyclic.

Geometric Topology;General Topology