  This work discusses the possible empirical verification of the geometrical
concept of homogeneity of the standard relativistic cosmology considering its
various definitions of distance. We study the physical consequences of the
distinction between the usual concept of spatial homogeneity (SH), as defined
by the Cosmological Principle, and the concept of observational homogeneity
(OH), arguing that OH is in principle falsifiable by means of astronomical
observations, whereas verifying SH is only possible indirectly. Simulated
counts of cosmological sources are produced by means of a generalized
number-distance expression that can be specialized to produce either the counts
of the Einstein-de Sitter (EdS) cosmology, which has SH by construction, or
other types of counts, which do, or do not, have OH by construction.
Expressions for observational volumes and differential densities are derived
with the various cosmological distance definitions in the EdS model. Simulated
counts that have OH by construction do not always exhibit SH features. The
reverse situation is also true. Besides, simulated counts with no OH features
at low redshift start showing OH characteristics at high redshift. The comoving
distance seems to be the only distance definition where both SH and OH appear
simultaneously. The results show that observations indicating possible lack of
OH do not necessarily falsify the standard Friedmannian cosmology, meaning that
this cosmology will not necessarily always produce observable homogeneous
densities. The general conclusion is that the use of different cosmological
distances in the characterization of the galaxy distribution lead to
significant ambiguities in reaching conclusions about the behavior of the
large-scale galaxy distribution in the Universe.
