Points, straight lines and planes are fundamental concept of geometry. Usually this entities are defined in a structure. We can easily define points in a vector space, or in a affine or projective space, but in this case what we define really is the structure of such spaces, of which points are elements and line are subspaces of subvarieties. So they are not defined for themselves, but only by means of the structure of which they are elements.
Also in a purely geometric approach, we know that they cannot be defined for themselves, but only by means of axioms that give meaning to such undefined primitives things. A classical way to do this is, e.g., by means of the axioms of Hilbert in his ''Foundations of Geometry'' where points, lines and planes are introduced as primitive undefined elements.
A way to eliminate the words ''point'' and ''line'' from the definitions is to use as structure a lattice, i.e. a partially ordered set in which any two elements have a least upper bound and a greatest lower bound. This is the approach of Menger and Birkhoff for the definition of projective geometries. Von Neumann showed that we can have lattices where there are no points or lines or planes (continous geometries) that generalize the projective geometries.
So it seams that for a good definition of the concept of ''point'' (and straight line) the more simple structure that we can use is a lattice. In some sense, points become well defined mathematical entities only if we have an order relation with suitable properties.
My question is: lattices are really the most simple structure that we need for define points, and what are exactly the properties that a structure must have for define points ( and straight lines and planes)?