Handbook of Finite Translation Planes (Pure and Applied Mathematics)
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This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines. A survey of linear topological geometries is given in Chapter 23 of the Handbook of incidence geometry. Such geometries are known as topological projective planes. A systematic study of these planes began in with a paper by Skornyakov.
What is Combinatorics? (A collection of quotes by Igor Pak)
Note that the rational numbers do not suffice to describe our intuitive notions of plane geometry and that some extension of the rational field is necessary. The approach to the topological properties of projective planes via ordering relations is not possible, however, for the planes coordinatized by the complex numbers , the quaternions or the octonion algebra. For details and other definitions of a dimension see  and the references given there, in particular Engelking  or Fedorchuk.
The lines of a compact topological plane with a 2-dimensional point space form a family of curves homeomorphic to a circle, and this fact characterizes these planes among the topological projective planes.
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The latter is known to imply the former Hessenberg . The theorem of Desargues expresses a kind of homogeneity of the plane.
This is possible due to the following basic theorem:. Topology of compact planes. Special aspects of 4-dimensional planes are treated in,  more recent results can be found in. Ahlfors and L. Sario, Riemann surfaces , Princeton Math. Press, Princeton, N. MR 22 Monographs, vol.
MR 41 MR 45 MR  P. Duren, Theory of spaces , Pure and Appl.
Journal of Algebraic Combinatorics (EMIS Electronic Version)
MR 42 MR 15 , MR a  W. Feller, An introduction to probability theory and its applications. II, Wiley, New York, MR 35 MR  R. Frankfurt and J. Rovnyak, Finite convolution operators , J. Math Anal. MR  J. Freeman, Volterra operators similar to , Trans.
Mer fra Norman Johnson
MR 33 MR  J. Ginsberg and D. Newman, Generators of certain radical algebras , J. Approximation Theory 3 , MR 36 ; 39 MR 36 ; 41 Kalisch, On similarity, reducing manifolds, and unitary equivalence of certain Volterra operators , Ann.
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- Handbook of Finite Translation Planes.
MR 19 , MR a  -, On isometric equivalence of certain Volterra operators , Proc. MR  -, On similarity invariants of certain operators in , Pacific J. Russian MR 32 MR 19 , ; 28 See Details. Product Spec Format: Hardcover.
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