In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set , it can be an element of the unit interval . Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor. Definition Menger and Birkhoff gave axioms for projective geometry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms. A continuous geometry is a lattice L with the following properties L is modular. L is complete. The lattice operations ∧, ∨ satisfy a certain continuity property, , where A is a directed set and if then , and the same condition with ∧ and ∨ reversed. Every element in L has a complement (not necessarily unique). A complement of an element a is an element b with , , where 0 and 1 are the minimal and maximal elements of L. L is irreducible: this means that the only elements with unique complements are 0 and 1. Examples Finite-dimensional complex projective space, or rather its set of linear subspaces, is a continuous geometry, with dimensions taking values in the discrete set The projections of a finite type II von Neumann algebra form a continuous geometry with dimensions taking values in the unit interval . showed that any orthocomplemented complete modular lattice is a continuous geometry. If V is a vector space over a field (or division ring) F, then there is a natural map from the lattice PG(V) of subspaces of V to the lattice of subspaces of that multiplies dimensions by 2. So we can take a direct limit of This has a dimension function taking values all dyadic rationals between 0 and 1. Its completion is a continuous geometry containing elements of every dimension in . This geometry was constructed by , and is called the continuous geometry over F Dimension This section summarizes some of the results of . These results are similar to, and were motivated by, von Neumann's work on projections in von Neumann algebras. Two elements a and b of L are called perspective, written , if they have a