Menger's connectivity theorem

The Verbindbarkeitssatz Menger is a mathematical theorem about a fundamental question of the theory of metric convex spaces and as such located in the transition area between the two mathematical areas topology and geometry . The sentence (like the concept of the metrically convex space) goes back to a work by the Austrian mathematician Karl Menger from 1928.

Formulation of the sentence

The sentence can be stated as follows:

A complete metric and at the same time metrically convex space is given . ${\displaystyle (X,d)}$

Then:

Between any two spatial points of any distance there is always a shortest connection in the sense that the associated real interval allows isometric embedding , which maps the real number to and the real number to . ${\displaystyle x,y\in X}$ ${\displaystyle A=d(x,y)>0}$ ${\displaystyle [0,A]}$ ${\displaystyle \phi \colon [0,A]\rightarrow (X,d)}$ ${\displaystyle 0}$ ${\displaystyle x=\phi (0)}$ ${\displaystyle A}$ ${\displaystyle y=\phi (A)}$

Related results

Another theorem is related to Menger's connectivity theorem, which is based on a similar question and which goes back to Stefan Mazurkiewicz :

In a topological space that is completely metrizable , connected and locally connected , there is always an open Jordan curve for every two different spatial points , which connects with . ${\displaystyle X}$ ${\displaystyle x,y\in X}$ ${\displaystyle \phi \colon [0,1]\rightarrow X}$ ${\displaystyle x=\phi (0)}$ ${\displaystyle y=\phi (1)}$

In connection with this - and no less also in connection with Menger's connectivity theorem - another sentence is worth mentioning that immediately follows and from Ákos Császár in his monograph General Topology as the theorem of Mazurkiewicz-Moore-Menger ( English Mazurkiewicz-Moore-Menger theorem ) referred to as. This sentence reads:

If a complete metric space is both connected and locally connected, then it is already connected arc-wise and locally connected arc-wise .

Notes on the proof of the theorem

Karl Menger derived the connectivity theorem using transfinite induction . In 1935, Nachman Aronszajn gave a proof without transfinite induction. Kazimierz Goebel and William A. Kirk have shown in their 1990 monograph Topics in Metric Fixed Point Theory that, based on Menger's original proof, one can produce a proof that uses a fixed point theorem instead of transfinite induction . As Goebel and Kirk explain, this fixed point theorem is a generalization of Banach's fixed point theorem and goes back to a publication by James Caristi from 1976. They call this generalization as a set of Caristi ( English Caristi's theorem ).

Caristi's theorem

The sentence says the following:

A complete metric space as well as a sub-continuous and also downwardly restricted real-valued function are given . ${\displaystyle (X,d)}$ ${\displaystyle \psi \colon (X,d)\rightarrow \mathbb {R} }$

Here is an arbitrary mapping that fulfills the following condition: ${\displaystyle f\colon \,X\to X}$

${\displaystyle d(x,f(x))\leq {\psi (x)-\psi (f(x))}}$

Then has a fixed point. ${\displaystyle f}$

See also

• Geodesics
• Geodetic metric space
• Jordan curve
• Peano rooms

literature

• N. Aronszajn : New proof of the continuity of the route of complete convex spaces . In: Results of a mathematical colloquium (Vienna) . tape 6 , 1935, pp. 45-56 .
• Leonard M. Blumenthal : Theory and Applications of Distance Geometry (=  Chelsea Scientific Books ). 2nd Edition. Chelsea Publishing Company, New York 1970, ISBN 0-8284-0242-6 ( MR0268781 ).
• James Caristi: Fixed point theorems for mappings satisfying inwardness conditions . In: Transactions of the American Mathematical Society . tape 215 , 1976, pp. 241-251 , doi : 10.2307 / 1999724 , JSTOR : 1999724 ( MR0394329 ).
• Ákos Császár : General Topology . 2nd Edition. Adam Hilger Ltd., Bristol 1978, ISBN 0-85274-275-4 ( MR0474162 ).
• Lutz Führer : General topology with applications . Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3 .
• Kazimierz Goebel, WA Kirk: Topics in Metric Fixed Point Theory (=  Cambridge Studies in Advanced Mathematics . Volume 28 ). Cambridge University Press, Cambridge 1990, ISBN 0-521-38289-0 ( MR1074005 ).
• Karl Menger: Research on general metrics . In: Mathematical Annals . tape 100 , 1928, pp. 75-163 ( uni-bielefeld.de ).
• RL Moore: On the foundations of plane analysis situs . In: Transactions of the American Mathematical Society . tape 17 , 1916, pp. 131-164 ( ams.org ).
• J. van Mill: The Infinite-dimensional Topology of Function Spaces (=  North-Holland Mathematical Library . Volume 64 ). North-Holland, Amsterdam (inter alia) 2002, ISBN 0-444-50557-1 .
• Willi Rinow : The inner geometry of the metric spaces (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 105 ). Springer Verlag, Berlin / Göttingen / Heidelberg 1961 ( MR0123969 ).

References and footnotes

1. ^ ^{A b c} Leonard M. Blumenthal: Theory and Applications of Distance Geometry. 1953, p. 32 ff, p. 41
2. ↑ ^{a b} Kazimierz Goebel, WA Kirk: Topics in Metric Fixed Point Theory. 1990, pp. 23-26
3. ↑ ^{a b} Willi Rinow: The inner geometry of the metric spaces. 1961, p. 146 ff., P. 148
4. ^ J. van Mill: The Infinite-dimensional Topology of Function Spaces. 2002, p. 55
5. ↑ Ákos Császár: General Topology. 1978, p. 428
6. ↑ The name "Moore" refers to Robert Lee Moore , who dealt with such connectivity issues in a work from 1916. See also the monograph General Topology with Applications. by Lutz Führer (Vieweg Verlag, Braunschweig 1977, p. 153 ff)!
7. ↑ See article "William Arthur Kirk" (English language Wikipedia) !
8. ↑ Goebel et al., Op. Cit., Pp. 9, 13, 24-25
9. ↑ In the Anglo-American literature the sentence is also called Caristi fixed-point theorem . See article "Caristi fixed-point theorem" (English language Wikipedia) !
10. ↑ Goebel et al., Op.cit., P. 13