  In this paper we study the nerves of two types of coverings of a sphere
$S^{d-1}$: (1) coverings by open hemispheres; (2) antipodal coverings by closed
hemispheres. In the first case, nerve theorem implies that the nerve is
homotopy equivalent to $S^{d-1}$. In the second case, we prove that the nerve
is homotopy equivalent to a wedge of $(2d-2)$-dimensional spheres. The number
of wedge summands equals the M\"{o}bius invariant of the geometric lattice (or
hyperplane arrangement) associated with the covering. This result explains some
observed large-scale phenomena in topological data analysis. We review the
particular case, when the coverings are centered in the root system $A_d$. In
this case the nerve of the covering by open hemispheres is the space of
directed acyclic graphs (DAGs), and the nerve of the covering by closed
hemispheres is the space of non-strongly connected directed graphs. The
homotopy types of these spaces were described by Bj\"{o}rner and Welker, and
the incarnation of these spaces appeared independently as "the poset of orders"
and "the poset of preorders" respectively in the works of Bouc. We study the
space of DAGs in terms of Gale and combinatorial Alexander dualities, and
propose how this space can be applied in automated machine learning.
