From Online Dictionary of Crystallography
A mathematical model of the space in which we live is the point space. Its elements are points. Objects in point space may be single points; finite sets of points like the centres of the atoms of a molecule; infinite discontinuous point sets like the centres of the atoms of an ideal crystal pattern; continuous point sets like straight lines, curves, planes, curved surfaces, etc.
Objects in point space are described by means of a coordinate system referred to point chosen as the origin O. An arbitrary point P is then described by its coordinates x, y, z.
The point space used in crystallography is a Euclidean space, i.e. an affine space where the scalar product is defined.
Crystal structures are described in point space. The vector space is a dual of the point space because to each pair of points in point space a vector in vector space can be associated.