  The basic hypothesis of a post-Copernican Cosmological theory is that {\em
all the points} of the Universe have to be essentially equivalent: this
hypothesis is required in order to avoid any privileged {\em observer}. This
assumption has been implemented by Einstein in the so-called Cosmological
Principles (CP): {\em all the positions} in the Universe have to be essentially
equivalent, so that the Universe is homogeneous. This situation implies also
the condition of spherical symmetry about every point, so that the Universe in
also Isotropic. There is a hidden assumption in the formulation of the CP in
regard to the hypothesis that all the points are equivalent In fact, the
condition that all the occupied points are statistically equivalent with
respect to their environment correspond to the property of of Local Isotropy.
It is generally believed that the Universe cannot be isotropic about every
point without being also homogeneous [1]. Actually Local Isotropy does not
necessarily implies homogeneity; in fact a topology theorem states that
homogeneity is implied by the condition of local isotropy together with {\em
the assumption of the analyticity or regularity} for the distribution of
matter. Up to the seventies analyticity was an obvious implicit assumption in
any physical problem. Recently however we have learned about intrinsically
irregular structures for which analyticity should be considered as a property
to be tested with appropriate analysis of experiment.
