  I provide a prescription to define space, at a given moment, for an arbitrary
observer in an arbitrary (sufficiently regular) curved space-time. This
prescription, based on synchronicity (simultaneity) arguments, defines a
foliation of space-time, which corresponds to a family of canonically
associated observers. It provides also a natural global reference frame (with
space and time coordinates) for the observer, in space-time (or rather in the
part of it which is causally connected to him), which remains Minkowskian along
his world-line. This definition intends to provide a basis for the problem of
quantization in curved space-time, and/or for non inertial observers.
  Application to Mikowski space-time illustrates clearly the fact that
different observers see different spaces. It allows, for instance, to define
space everywhere without ambiguity, for the Langevin observer (involved in the
Langevin pseudoparadox of twins). Applied to the Rindler observer (with uniform
acceleration) it leads to the Rindler coordinates, whose choice is so justified
with a physical basis. This leads to an interpretation of the Unruh effect, as
due to the observer's dependence of the definition of space (and time). This
prescription is also applied in cosmology, for inertial observers in the
Friedmann - Lemaitre models: space for the observer appears to differ from the
hypersurfaces of homogeneity, which do not obey the simultaneity requirement. I
work out two examples: the Einstein - de Sitter model, in which space, for an
inertial observer, is not flat nor homogeneous, and the de Sitter case.
