Topological projective plane

A topological projective level is a projective level on whose point and line set a topology is explained in such a way that the formation of the intersection of two lines and the formation of the connecting lines are continuous operations. In addition, the topology of the point set should not be indiscreet . Even these weak preconditions result in very strong separation properties .

history

In the early 1930s has Andrey Kolmogorov the notion of topological level used to be below described to prove important result. Apparently, this was hardly noticed at the time outside the Soviet Union.

Coming from the basics of geometry and inspired by ideas from the theory of arranged and topological bodies , the first broader studies of topological projective planes arose in the 1950s (for example by Oswald Wyler and LA Skornjakow). An extensive theory of these levels was then developed by Helmut Salzmann and his students.

In the 1960s, the concept was applied to closure sets by Sibylla Crampe and others .

Definitions and spellings

It is a projective plane , are systems of subsets of the set of points or  set of lines . The following spellings are agreed for connection and section: ${\displaystyle \mathbb {P} =({\mathcal {P}},{\mathcal {G}},\operatorname {I} )}$ ${\displaystyle {\mathcal {P}}\cap {\mathcal {G}}=\emptyset }$ ${\displaystyle T_{\mathcal {P}}\subseteq 2^{\mathcal {P}},T_{\mathcal {G}}\subseteq 2^{\mathcal {G}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {G}}}$

${\displaystyle \sqcup :{\mathcal {P}}\times {\mathcal {P}}\setminus \{(P,P)\in {\mathcal {P}}\times {\mathcal {P}}\}\rightarrow {\mathcal {G}}\colon (P,Q)\mapsto P\sqcup Q:=g\Leftrightarrow (P\operatorname {I} g)\land (Q\operatorname {I} g)}$

${\displaystyle \sqcap :{\mathcal {G}}\times {\mathcal {G}}\setminus \{(g,g)\in {\mathcal {G}}\times {\mathcal {G}}\}\rightarrow {\mathcal {P}}\colon (g,h)\mapsto g\sqcap h:=P\Leftrightarrow (P\operatorname {I} g)\land (P\operatorname {I} h)}$

Then the topological projective plane is called if the following conditions are met: ${\displaystyle (\mathbb {P} ,T_{\mathcal {P}},T_{\mathcal {G}})}$

1. ${\displaystyle ({\mathcal {P}},T_{\mathcal {P}})}$and are topological spaces . ${\displaystyle ({\mathcal {G}},T_{\mathcal {G}})}$
2. The operation is continuous with respect to the product topology on and ${\displaystyle \sqcup }$ ${\displaystyle ({\mathcal {P}},T_{\mathcal {P}})}$
3. The operation is constantly based on the product topology . ${\displaystyle \sqcap }$ ${\displaystyle ({\mathcal {G}},T_{\mathcal {G}})}$
4. The topology on the point space is not the indiscreet topology , that is: There is an open point set apart from the empty set and . ${\displaystyle {\mathcal {P}}}$

For the environment filter mapping one writes for points as well as straight lines , so one defines: ${\displaystyle \Omega }$

${\displaystyle \Omega \colon ({\mathcal {P}}\cup {\mathcal {G}})\rightarrow {T_{\mathcal {P}}}\cup {T_{\mathcal {G}}}\colon \quad \xi \mapsto \Omega (\xi )=\{U|(\xi \in U)\land \left((U\in T_{\mathcal {P}})\lor (U\in T_{\mathcal {G}})\right)\}}$

The second condition can then be formulated in more detail as follows: For different points and every open set of straight lines there are open sets of points , so that it always follows. The third condition can be formulated in two ways. The fact that the fourth dual statement applies in every topological level follows from much stronger statements that are presented in the next section. ${\displaystyle A,B}$ ${\displaystyle U_{0}\in \Omega (A\sqcup B)}$ ${\displaystyle U_{1}\in \Omega (A),U_{2}\in \Omega (B)}$ ${\displaystyle X\in U_{1},Y\in U_{2},X\neq Y}$ ${\displaystyle (X\sqcup Y)\in U_{0}}$

properties

Perspective assignment of a row of points to a cluster. This mapping is continuous in a topological projective level including its inversion.

In a topological projective plane:

• The perspective assignment of a row of points (the set of points on a straight line ) to a tuft (the set of straight lines through a point ) is a homeomorphism . ${\displaystyle a\in {\mathcal {G}}}$ ${\displaystyle Z\in {\mathcal {P}};\;\neg (Z\operatorname {I} a)}$
• Any two rows of points are homeomorphic to one another.
• The topology of the plane is discrete if and only if the topology induced on a row of points is discrete.
• The point set of an affine plane is homeomorphic to the topological product of an affine point series with itself. ${\displaystyle A_{g}={\mathcal {P}}\setminus \{P|P\operatorname {I} g\;\};\;(g\in {\mathcal {G}})}$
• Every row of points and every single-element set of points is closed (in the topological space of points).

So the point set is a - or Kolmogoroff space . It also follows from this that the topology must be discrete if the plane is finite. ${\displaystyle T_{0}}$

• The set of points and the set of lines of the plane are regular spaces .

In this way, since they also fulfill, they also fulfill and are Hausdorff rooms . ${\displaystyle T_{0}}$ ${\displaystyle T_{1},T_{2},T_{3}}$

• The point space of the plane is either contiguous or nowhere contiguous . The same alternative applies to a point series, affine point series and the point set of an affine plane, each of these subspaces is connected if and only if it is the whole point space.
• The local compactness of the entire point set, a point series, an affine point series and that of the point set of an affine plane are equivalent: Either all these spaces are locally compact or none.
• If the point set of the level is locally compact, then it has a countable environment base .

Conclusions from topological properties for the coordinate area

• (A theorem by Kolmogorow) A desargue , locally compact, connected topological projective plane is too continuously isomorphic, where the oblique body is either the field of real numbers , the field of complex numbers or the quaternion oblique body . ${\displaystyle \mathbb {P} ^{2}(S)}$ ${\displaystyle S}$
• If in a topological projective plane one (and thus each) of the rows of points is a manifold , then it is a Moufang plane and continuously isomorphic to , whereby one of the oblique fields from the previous statement or the alternative field is the (real) octonion . ${\displaystyle \mathbb {P} ^{2}(C)}$ ${\displaystyle C}$

In both statements, the topology means that which is induced by its respective coordinate space as a real vector space with its usual topology. ${\displaystyle \mathbb {P} ^{2}(C)}$

Examples

• Each projective plane with the discrete topology on its set of points and lines becomes a topological projective plane. As noted above, this is the only possible topology for finite levels.
• The projective level above the rational numbers , together with the topology induced by the order topology of the affine level and transferable to its projective closure, is an example of a topological level that is nowhere connected but not discrete. ${\displaystyle \mathbb {P} ^{2}(\mathbb {Q} )}$ ${\displaystyle \mathbb {Q} }$

literature

Original article

• Sibylla Crampe : Closing sentences in projective planes and dense partial planes . In: Archives of Mathematics . tape 11 , no. 1 . Birhäuser, December 1, 1960, ISSN  1420-8938 , p. 136-145 , doi : 10.1007 / BF01236921 .
• Andrei Nikolajewitsch Kolmogorow : To the foundation of the projective geometry . In: Ann. of Math . tape 33 , 1932, pp. 175-176 .
• Helmut Salzmann: About the connection in topological projective levels . In: Mathematical Journal . tape 61 , no. 1 . Springer, 1955, ISSN  1432-1823 , p. 489-494 , doi : 10.1007 / BF01181361 .
• Helmut Salzmann: Topological projective levels . In: Mathematical Journal . tape 67 , no. 1 . Springer, December 1957, ISSN  1432-1823 , p. 436-466 , doi : 10.1007 / BF01258875 .
• Helmut Salzmann: Topological planes . In: Advances in Mathematics . tape 2 , 1967, p. 1-60 .
• LA Skornjakow: Topological Projective Planes . In: Trudy Moskov. Math. Obšč. tape 3 , 1954, pp. 347-373 .
• Oswald Wyler: Order and topology in projective planes . In: American Journal of Mathematics . tape 74 , no. 3 . Johns Hopkins University Press, 1952, pp. 656-666 , JSTOR : 2372268 .

Textbooks

• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 , 10. Topological levels .
• Sibylla Prieß-Crampe : Arranged structures . Groups, bodies, projective levels (=  results of mathematics and its border areas . Volume 98 ). Springer, Berlin / Heidelberg / New York 1983, ISBN 3-540-11646-X , V §2 Basic Concepts of Topological Projective Planes , p. 189-217 .

References and comments

1. ↑ ^{a b} Kolmogorow (1932)
2. ^ VM Tikhomirow: Selected Works of AN Kolmogorov . Mathematics and Mechanics. tape I . Kluwer Academic Publishers, Dordrecht / Boston / London 1992 ( book review link.springer.com [PDF; accessed August 10, 2013]).
3. ↑ Wyler (1952)
4. ↑ ^{a b c} Skornjakow (1954)
5. ^ Prieß-Crampe (1983), foreword
6. ↑ for example Crampe (1960)
7. ↑ This formal condition, which says that no point equals a straight line, is usually assumed for projective planes anyway. It is necessary here so that environmental filters can be represented in a somewhat simplified manner. According to Pickert (1975), p. 361
8. ↑ There is no common notation for these operations. The connection symbols chosen here are the abstract ones for a union and express the duality of the two operations well
9. ↑ after Pickert (1975), p. 361f
10. ↑ Prieß-Crampe (1983), V §2 sentence 1
11. ↑ Prieß-Crampe (1983), V §2 sentence 6
12. ↑ ^{a b c d} Salzmann (1955)
13. ↑ Prieß-Crampe (1983), V §2 sentence 5
14. ↑ Prieß-Crampe (1983), V §2 sentence 4
15. ↑ Prieß-Crampe (1983), V §2 sentence 11
16. ↑ Prieß-Crampe (1983), V §2 sentence 12
17. ^ Salzmann (1957)
18. ↑ Prieß-Crampe (1983), V §2 sentence 16
19. ↑ Pickert (1975) 10.1. Sentence 9
20. ↑ Salzman (1967), Th.7.12
21. ↑ Pickert (1975), p. 267