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  "body": "\n## Summary\n\nA continuous geometry is a complete, complemented, [[modular lattice|QX7WZ92QAK]] in which meets and joins are continuous along chains, and which is irreducible.\nJohn von Neumann introduced continuous geometries as analogues of projective geometries.\nIn them the dimension can take every value in $[0, 1]$, not only integer values.\nThe lattice of projections of a type II$_1$ factor is the motivating example.\nEvery continuous geometry carries a dimension function, and von Neumann related these lattices to [[von Neumann regular rings|DXX7GNY3HE]].\n\n## Explanations\n\n### L1 human\n\nIn a projective geometry, subspaces have dimensions $0, 1, 2, \\dots, n$, and after rescaling by $n$ these are the numbers $0, 1/n, \\dots, 1$.\nVon Neumann asked for a geometry whose subspaces can have any real dimension between $0$ and $1$.\nSuch a \"geometry\" has no points or lines, only subspaces, so it is described by the lattice of its subspaces.\nThe axioms ask that this lattice behaves like a subspace lattice and that limits of increasing or decreasing families behave well.\n",
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