Complete Lattice: Continuous Geometry
by John von Neumann
In his work on rings of operators in Hilbert space, John von Neumann discovered a new mathematical structure that resembled the lattice system Ln. In characterizing its properties, von Neumann founded the field of continuous geometry. This book, based on von Neumann’s lecture notes, begins with the development of the axioms of continuous geometry, dimension theory, and–for the irreducible case–the function D(a). The properties of regular rings are then discussed, and a variety of results are presented for lattices that are continuous geometries, for which irreducibility is not assumed. For students and researchers interested in ring theory or projective geometries, this book is required reading.
Complete Lattice: Finite topological space
“Every finite bounded lattice is complete since the meet or join of any family of elements can always be reduced to a meet or join of two elements. It follows that in a finite topological space the union or intersection of an arbitrary family of open sets (resp. closed sets) is open (resp. closed).”