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Topological projective spaces

Topological projective spaces Abh. Math. Sem. Univ. Hamburg 62 (1992), 1-9 By R. KiJ'I-INE and R. L6WEN After the first sketches by KOLMOGOROV [3] and PONTRJAGIN [8] w Beispiel 48, various approaches to a theory of finite dimensional desarguesian topological projective spaces have been published by LENZ [5], MISFELD [7], S6RENSEN [9], DOIGNON [1], SZAMBIEN [10] and GROH [2]. Given a topological field, one wishes to construct a topology on any projective space associated with it, such that geometric operations become continuous in some suitable sence ('existence theorem'); moreover, one hopes that the continuity properties are strong enough to characterize this topology ('representation theorem'). The choice of continuity condition (i.e. the definition of the notion 'topological projective space') is by no means obvious. LENZ'S definition is more general than the other ones, and does not lead to a representation theorem unless compactness is required. The other approaches do yield representation theorems and hence are equivalent in some sense. The original one by MISF~LD gives the strongest existence theorem since his continuity conditions involve the whole subspace lattice. His notion was extended by GROH SO as to encompass hyperbolic spaces and other open subgeometries. However, MISFELD'S constructions and proofs are very complicated. A http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (8)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941614
Publisher site
See Article on Publisher Site

Abstract

Abh. Math. Sem. Univ. Hamburg 62 (1992), 1-9 By R. KiJ'I-INE and R. L6WEN After the first sketches by KOLMOGOROV [3] and PONTRJAGIN [8] w Beispiel 48, various approaches to a theory of finite dimensional desarguesian topological projective spaces have been published by LENZ [5], MISFELD [7], S6RENSEN [9], DOIGNON [1], SZAMBIEN [10] and GROH [2]. Given a topological field, one wishes to construct a topology on any projective space associated with it, such that geometric operations become continuous in some suitable sence ('existence theorem'); moreover, one hopes that the continuity properties are strong enough to characterize this topology ('representation theorem'). The choice of continuity condition (i.e. the definition of the notion 'topological projective space') is by no means obvious. LENZ'S definition is more general than the other ones, and does not lead to a representation theorem unless compactness is required. The other approaches do yield representation theorems and hence are equivalent in some sense. The original one by MISF~LD gives the strongest existence theorem since his continuity conditions involve the whole subspace lattice. His notion was extended by GROH SO as to encompass hyperbolic spaces and other open subgeometries. However, MISFELD'S constructions and proofs are very complicated. A

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 28, 2008

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