  Traditionally, phase transitions are defined in the thermodynamic limit only.
We propose a new formulation of equilibrium thermo-dynamics that is based
entirely on mechanics and reflects just the {\em geometry and topology} of the
N-body phase-space as function of the conserved quantities, energy, particle
number and others. This allows to define thermo-statistics {\em without the use
of the thermodynamic limit}, to apply it to ``Small'' systems as well and to
define phase transitions unambiguously also there. ``Small'' systems are
systems where the linear dimension is of the characteristic range of the
interaction between the particles. Also astrophysical systems are ``Small'' in
this sense. Boltzmann defines the entropy as the logarithm of the area
$W(E,N)=e^{S(E,N)}$ of the surface in the mechanical N-body phase space at
total energy E. The topology of S(E,N) or more precisely, of the curvature
determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial
N^2-(\partial^2S/\partial E\partial N)^2$ allows the classification of phase
transitions {\em without taking the thermodynamic limit}. The topology gives
further a simple and transparent definition of the {\em order parameter.}
Attention: Boltzmann's entropy S(E) as defined here is different from the
information entropy and can even be non-extensive and convex.
