The definition

Given an arbitrary linear space S = (p, ℒ), there is a natural way of introducing further structure on S by distinguishing subsets of p with the property that any element of ℒ containing at least two points of such a subset must be wholly contained within the subset. We call any subset of p with the above property, along with the induced lines, a subspace of 5. Clearly φ and S themselves are subspaces. Moreover, the intersection of any set of subspaces is again a subspace (the empty intersection being S), and so a closure space is induced on S, the closure of any set X of points of p being the intersection of all subspaces containing X.

There are various ways in which a ‘dimension’ can now be assigned to subspaces. In any case, we normally wish to assign dimensions 0 and 1 to the points and lines respectively, and require that V ⊂ W implies that the dimension of V be less than or equal to the dimension of W. We shall make the following definition.