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.

21 pages, 6 figures. Astronomy & Astrophysics, in press