## Convergence of Automorphisms of Compact Projective Planes (0)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Kummetz_convergenceof,

author = {Ralph Kummetz},

title = {Convergence of Automorphisms of Compact Projective Planes},

year = {}

}

### OpenURL

### Abstract

Introduction Convergence and continuity properties of homomorphisms play an important role in the theory of topological projective planes. Grundhofer [8] showed that the set \Sigma of all automorphisms of a compact projective plane is a locally compact transformation group with respect to the topology of uniform convergence; for the special case of compact connected projective planes see also Salzmann [21]. With regards to classification, compact connected projective planes have been successfully investigated by studying their automorphism group, see Salzmann, Betten, Grundhofer, Hahl, Lowen, and Stroppel [22] for a detailed exposition. Salzmann [20] proved that if \Pi is a 2-dimensional compact projective plane, then on \Sigma the topology of pointwise convergence coincides with the topology of uniform convergence. He also showed ([19]) that any homomorphism between 2-dimensional compact projective planes is in fact a homeomorphism. Grundhofer [9] characterized the continuity of non-

### Citations

241 |
General Topology
- Engelking
- 1989
(Show Context)
Citation Context ...notation. For the basic concepts of plane projective geometry, the reader is referred to Pickert [13] or Hughes/Piper [12]; for topological notions see e.g. Bourbaki [2], [3], Dugundji [5], Engelking =-=[6]-=-. Four points of a projective plane are called a quadrangle if any three of them are not collinear. A topological projective plane \Pi = (P; L;sP ;sL ) is a projective plane (P; L) with neither indisc... |

46 |
Projective planes
- Hughes, Piper
- 1973
(Show Context)
Citation Context ...nt. (Observe the analogy to homomorphisms of topological groups.) Let us fix some notation. For the basic concepts of plane projective geometry, the reader is referred to Pickert [13] or Hughes/Piper =-=[12]-=-; for topological notions see e.g. Bourbaki [2], [3], Dugundji [5], Engelking [6]. Four points of a projective plane are called a quadrangle if any three of them are not collinear. A topological proje... |

33 |
Elements of Mathematics
- Bourbaki
- 2008
(Show Context)
Citation Context ...cal groups.) Let us fix some notation. For the basic concepts of plane projective geometry, the reader is referred to Pickert [13] or Hughes/Piper [12]; for topological notions see e.g. Bourbaki [2], =-=[3]-=-, Dugundji [5], Engelking [6]. Four points of a projective plane are called a quadrangle if any three of them are not collinear. A topological projective plane \Pi = (P; L;sP ;sL ) is a projective pla... |

21 |
Uniform structures in topological groups and their quotients
- Roelcke, Dierolf
- 1981
(Show Context)
Citation Context ...IN converges uniformly to ff. Since \Sigma, endowed with the topology of uniform convergence, is a locally compact group, it is complete with respect to its right uniformity, cf. e.g. Roelcke/Dierolf =-=[16]-=-, Proposition 8.8. Let d sup be the supremum metric on P P and d sup j \Sigma its restriction on \Sigma. One easily sees that d sup j \Sigma is right invariant, i.e. d sup (oe 1 ; oe 2 ) = d sup (oe 1... |

18 |
Elements of Mathematics. General Topology, Part 1, Hermann
- Bourbaki
- 1966
(Show Context)
Citation Context ...ological groups.) Let us fix some notation. For the basic concepts of plane projective geometry, the reader is referred to Pickert [13] or Hughes/Piper [12]; for topological notions see e.g. Bourbaki =-=[2]-=-, [3], Dugundji [5], Engelking [6]. Four points of a projective plane are called a quadrangle if any three of them are not collinear. A topological projective plane \Pi = (P; L;sP ;sL ) is a projectiv... |

17 |
Projektive Ebenen
- Pickert
- 1955
(Show Context)
Citation Context ...ontinuous at some point. (Observe the analogy to homomorphisms of topological groups.) Let us fix some notation. For the basic concepts of plane projective geometry, the reader is referred to Pickert =-=[13]-=- or Hughes/Piper [12]; for topological notions see e.g. Bourbaki [2], [3], Dugundji [5], Engelking [6]. Four points of a projective plane are called a quadrangle if any three of them are not collinear... |

11 |
Collinearity-preserving functions between Desarguesian planes
- Carter, Vogt
- 1980
(Show Context)
Citation Context ... a point in P , then we write L p for the set of all lines passing through p. A mapping ff : P ! P 0 is called a lineation if ff maps collinear points of P to collinear points of P 0 (cf. Carter/Vogt =-=[4]-=-). A lineation ff is called a homomorphism if the image of ff contains a quadrangle. Let ff : P ! P 0 be a homomorphism. The dual mapping of ff is defined by ff : ( L \Gamma! L 0 L = asb 7\Gamma! ff(a... |

6 |
Topology in lattices
- Frink
- 1942
(Show Context)
Citation Context ... the definition of the continuous convergence as it can be found 2 e.g. in Arens/Dugundji [1], cf. also Poppe [14] (Chapter 2.2). We "localize" this concept of convergence to single points, =-=see Frink [7]-=-, x15, and for a special case also Hahn [11], x28.9. We note that we deal with nets and not just with sequences of mappings, since nets are needed for the proof of Corollary (3.2). Let (X;s) and (Y; A... |

4 |
Topologies for function spaces
- Arens, Dugundji
- 1951
(Show Context)
Citation Context ...s f i : X ! Y to a continuous mapping f : X ! Y , where X is a compact space and Y is a uniform space. We give the definition of the continuous convergence as it can be found 2 e.g. in Arens/Dugundji =-=[1], cf. also-=- Poppe [14] (Chapter 2.2). We "localize" this concept of convergence to single points, see Frink [7], x15, and for a special case also Hahn [11], x28.9. We note that we deal with nets and no... |

4 |
Homomorphismen topologischer projektiver Ebenen
- Salzmann
- 1959
(Show Context)
Citation Context ... Salzmann [20] proved that if \Pi is a 2-dimensional compact projective plane, then on \Sigma the topology of pointwise convergence coincides with the topology of uniform convergence. He also showed (=-=[19]-=-) that any homomorphism between 2-dimensional compact projective planes is in fact a homeomorphism. Grundhofer [9] characterized the continuity of non-injective epimorphisms ff between arbitrary topol... |

3 |
On restrictions of automorphism groups of compact projective planes to subplanes
- Grundhofer
- 1992
(Show Context)
Citation Context ... if and only if it is continuous at some point. 1 Introduction Convergence and continuity properties of homomorphisms play an important role in the theory of topological projective planes. Grundhofer =-=[8]-=- showed that the set \Sigma of all automorphisms of a compact projective plane is a locally compact transformation group with respect to the topology of uniform convergence; for the special case of co... |

3 |
Compact Disconnected Planes, Inverse Limits and Homomorphisms
- Grundhofer
- 1988
(Show Context)
Citation Context ...wise convergence coincides with the topology of uniform convergence. He also showed ([19]) that any homomorphism between 2-dimensional compact projective planes is in fact a homeomorphism. Grundhofer =-=[9]-=- characterized the continuity of non-injective epimorphisms ff between arbitrary topological projective planes by considering places which belong to ff. In this note we will investigate under which co... |

3 |
Topologische projektive Ebenen
- Salzmann
- 1957
(Show Context)
Citation Context ...L;sL ) are compact. See Salzmann et al. [22] for a thorough study of compact projective planes. Throughout the remainder of this section let \Pi be compact. Since (P;sP ) is then metrizable (Salzmann =-=[18]-=-, x3; Salzmann et al. [22], Theorem 41.8), the topology of uniform convergence on the function space P P := f f j f : P ! P g is induced by the supremum metric. Recall that on the subset of all contin... |

2 |
Compactness in General Function Spaces. VEB Deutscher Verlag der Wissenschaften
- Poppe
- 1974
(Show Context)
Citation Context ...ontinuous mapping f : X ! Y , where X is a compact space and Y is a uniform space. We give the definition of the continuous convergence as it can be found 2 e.g. in Arens/Dugundji [1], cf. also Poppe =-=[14] (Chapter -=-2.2). We "localize" this concept of convergence to single points, see Frink [7], x15, and for a special case also Hahn [11], x28.9. We note that we deal with nets and not just with sequences... |

2 |
B.: Mengentheoretische Topologie
- Querenburg
- 1979
(Show Context)
Citation Context ...oint of continuous convergence of (f n ) n2IN and f g. Then X n T is meagre in X. If, in addition, (X;s) is a Baire space, then T is dense and thick in X. Remark: The proof is similar to the proof of =-=[15]-=-, Satz 13.32. This theorem states that if f : X ! IR is the pointwise limit of a sequence of continuous functions and S is the set of all points at which f is continuous, then XnS is meagre in X. Proo... |

2 |
Homogene kompakte projektive Ebenen
- Salzmann
- 1975
(Show Context)
Citation Context ...a compact projective plane is a locally compact transformation group with respect to the topology of uniform convergence; for the special case of compact connected projective planes see also Salzmann =-=[21]-=-. With regards to classification, compact connected projective planes have been successfully investigated by studying their automorphism group, see Salzmann, Betten, Grundhofer, Hahl, Lowen, and Strop... |

2 |
Endomorphisms of Stable Planes. Sem. Sophus Lie 2
- Stroppel
- 1992
(Show Context)
Citation Context ...is a complete metric subspace of (P P ; d sup ) and ff 2 \Sigma. (We just remark here that if \Pi were furthermore connected, then the assertion 6 "ff 2 \Sigma" would follow immediately from=-= Stroppel [23]-=-, Corollary 9.) Conversely, let ff contain a quadrangle in its image. Since (P;sP ) is a compact metrizable space, we find a point of continuous convergence of (oe n ) n2IN and ff by Proposition (2.3)... |

1 |
Reelle Funktionen. Erster Teil: Punktfunktionen
- Hahn
- 1948
(Show Context)
Citation Context ...ce as it can be found 2 e.g. in Arens/Dugundji [1], cf. also Poppe [14] (Chapter 2.2). We "localize" this concept of convergence to single points, see Frink [7], x15, and for a special case =-=also Hahn [11]-=-, x28.9. We note that we deal with nets and not just with sequences of mappings, since nets are needed for the proof of Corollary (3.2). Let (X;s) and (Y; AE) be topological spaces, x 2 X, (f i ) i2I ... |

1 |
den Zusammenhang in topologischen projektiven Ebenen
- Salzmann
- 1955
(Show Context)
Citation Context ... L 0 ), see again [4]. From now on let \Pi = (P; L;sP ;sL ) and \Pi 0 = (P 0 ; L 0 ;sP 0 ;sL 0 ) be topological projective planes. Recall that the mentioned topologies are regular Hausdorff (Salzmann =-=[17]-=-; Salzmann et al. [22], Proposition 41.4). (3.1) Proposition: Let ff : P ! P 0 be a mapping with a quadrangle in its image and (ff i ) i2I a net of lineations from P to P 0 . Assume that (ff i ) i2I c... |

1 |
Kompakte zweidimensionale projektive
- Salzmann
- 1962
(Show Context)
Citation Context ...nected projective planes have been successfully investigated by studying their automorphism group, see Salzmann, Betten, Grundhofer, Hahl, Lowen, and Stroppel [22] for a detailed exposition. Salzmann =-=[20]-=- proved that if \Pi is a 2-dimensional compact projective plane, then on \Sigma the topology of pointwise convergence coincides with the topology of uniform convergence. He also showed ([19]) that any... |