---
tag: 5CET1KWWWH
kind: concept
name: Continuous geometry
msc: [06C20, 51A05]
specializes: [QX7WZ92QAK]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_geometry
same_as:
  wikidata: Q5165479
  wikipedia: Continuous geometry
---

## Summary

A continuous geometry is a complete, complemented, [[modular lattice|QX7WZ92QAK]] in which meets and joins are continuous along chains, and which is irreducible.
John von Neumann introduced continuous geometries as analogues of projective geometries.
In them the dimension can take every value in $[0, 1]$, not only integer values.
The lattice of projections of a type II$_1$ factor is the motivating example.
Every continuous geometry carries a dimension function, and von Neumann related these lattices to [[von Neumann regular rings|DXX7GNY3HE]].

## Explanations

### L1 human

In a projective geometry, subspaces have dimensions $0, 1, 2, \dots, n$, and after rescaling by $n$ these are the numbers $0, 1/n, \dots, 1$.
Von Neumann asked for a geometry whose subspaces can have any real dimension between $0$ and $1$.
Such a "geometry" has no points or lines, only subspaces, so it is described by the lattice of its subspaces.
The axioms ask that this lattice behaves like a subspace lattice and that limits of increasing or decreasing families behave well.