  It is proved that any static system that is spacetime-geodesically complete
at infinity, and whose spacelike-topology outside a compact set is that of R^3
minus a ball, is asymptotically flat. The matter is assumed compactly supported
and no energy condition is required. A similar (though stronger) result applies
to black holes too. This allows us to state a large generalisation of the
uniqueness of the Schwarzschild solution not requiring asymptotic flatness. The
Korotkin-Nicolai static black-hole shows that, for the given generalisation, no
further flexibility in the hypothesis is possible.
