Jordan-Brouwer decomposition theorem

The Jordan-Brouwer decomposition theorem is a theorem of topology , which generalizes the Jordanian curve theorem from two to dimensions. It goes back to the French mathematician Camille Jordan and the Dutch mathematician Luitzen Egbertus Jan Brouwer . In German-language literature, the sentence can also be found as a separation sentence from Jordan-Brouwer or as a decomposition sentence from Jordan-Brouwer-Alexander . The latter naming takes into account the work that the American mathematician James Waddell Alexander contributed to this topic. ${\displaystyle n}$

statement

The Jordan-Brouwer decomposition theorem (in today's version) is:

Let and be homeomorphic compact subsets of the . Then the complements and have the same number of path components . ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}\setminus K}$ ${\displaystyle \mathbb {R} ^{n}\setminus L}$

More specific formulation:

When a compact and coherent hypersurface of , then there is the complement of , that is , from two open connected sets, the "inside" and "outside" . It is the closure of the interior, that is , a compact manifold with the edge , ie . ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}\setminus X}$ ${\displaystyle D_{1}}$ ${\displaystyle D_{0}}$ ${\displaystyle {\overline {D}}_{1}}$ ${\displaystyle X}$ ${\displaystyle \partial {\overline {D}}_{1}=X}$

Inferences

In addition to the Jordan curve theorem , the Jordan-Brouwer decomposition theorem entails further theorems of the topology of the n-dimensional Euclidean space . This gives an indication of its fundamental importance.

Theorem of the invariance of open sets

Let be an open subset of and an injective continuous map . Then there is also an open subset of and even a homeomorphism . ${\displaystyle U}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon U\to \mathbb {R} ^{n}}$ ${\displaystyle f(U)\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon \,U\to f(U)}$

In the German-language literature, the sentence is also quoted under the similar keyword invariance of the open set .

Since the connection or the path connection is always retained under continuous mapping, the following invariance theorem immediately results as a corollary.

Theorem of the invariance of the area

Let be a domain of and an injective continuous map . Then there is also a domain of and even a homeomorphism . ${\displaystyle U}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon \,U\to \mathbb {R} ^{n}}$ ${\displaystyle f(U)\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon \,U\to f(U)}$

In the English-language literature, this sentence can be found under the keyword Invariance of domain .

Theorem of the invariance of dimension

Let be an open subset of and be an open subset of . If and are homeomorphic, then . ${\displaystyle \emptyset \neq U}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \emptyset \neq V}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle n=m}$

In particular and for homeomorphic never. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle n\neq m}$

In the English-language literature, this sentence can be found under the keyword Invariance of dimension .

In 1879, Eugen Netto proved that the bijective mapping of the unit interval onto the unit square by Georg Cantor cannot be continuous.

Meaning of the sentences, derivation, historical

The meaning of the decomposition theorem and the invariance theorems (and thus the meaning of Brouwer's achievement) is based not least on the contribution to clarifying the question of the nature of the dimension of space, which has been under discussion since Georg Cantor . Cantor had in correspondence with Richard Dedekind shown that and , thus , and then all the same thickness have, so that and     for   bijektiv be mapped to each other. However, it was suggested (following Dedekind) that no such bijection could be a homeomorphism . Brouwer was the first to prove this. It is no less significant that Brouwer introduced new, fruitful methods into topology for the derivation of his theorems. In particular, the degree of mapping (English degree ) for continuous functions goes back to Brouwer, who has subsequently proven to be a very useful tool. ${\displaystyle [0,1]\,}$ ${\displaystyle [0,1]\times [0,1]}$ ${\displaystyle \mathbb {R} \,}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}\,}$ ${\displaystyle \mathbb {R} ^{m}\,}$ ${\displaystyle n\neq m\,}$  

James Waddell Alexander was able to show in 1922 that another approach is possible in addition to Brouwer's approach. He proved that his duality theorem entails the decomposition theorem . The theorems of the invariance of open sets , the invariance of the area and the invariance of the dimension can already be derived within the framework of the singular homology theory . As Emanuel Sperner was able to show in 1928, the latter can also be proven using elementary combinatorial aids alone.

literature

Original work

• James W. Alexander : A proof and extension of the Jordan-Brouwer separation theorem . In: Transactions of the American Mathematical Society . tape 23 , no. 4 , 1922, pp. 333-349 , doi : 10.2307 / 1988883 .
• Luitzen EJ Brouwer : Proof of the invariance of the dimension number . In: Mathematical Annals . tape 70 , 1911, pp. 161-165 ( digitized version ).
• Luitzen EJ Brouwer: About the mapping of manifolds. In: Mathematical Annals. Volume 71, 1912, pp. 97–115, ( digitized ; corrections in: Mathematische Annalen. Volume 71, 1912, p. 598, doi : 10.1007 / BF01456812 and Volume 82, 1921, p. 286, doi : 10.1007 / BF01498670 ) .
• Luitzen EJ Brouwer: Proof of the invariance of the n-dimensional domain . In: Mathematical Annals . tape 71 , 1912, pp. 305-313 ( digitized version ).
• Luitzen EJ Brouwer: Proof of Jordan theorem for n-dimensional space . In: Mathematical Annals . tape 71 , 1912, pp. 314-319 ( digitized version ).
• Luitzen EJ Brouwer: On the invariance of the n-dimensional area . In: Mathematical Annals . tape 72 , 1912, pp. 55-56 ( digitized version ).
• Egbert Harzheim : Generalization of the decomposition theorem by Jordan-Brouwer-Alexander to products of linearly ordered continuities . In: Archives of Mathematics . tape 46 , no. 3 , 1986, pp. 271-274 , doi : 10.1007 / BF01194195 .
• Emanuel Sperner : New proof for the invariance of the dimensional number and the area . In: Treatises from the Mathematical Seminar of the University of Hamburg . tape 6 , 1928, pp. 265-272 , doi : 10.1007 / BF02940617 .

Monographs

• Lutz Führer : General topology with applications . Vieweg, Braunschweig 1977, ISBN 3-528-03059-3 .
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Karl Heinz Mayer: Algebraic Topology . Birkhäuser, Basel et al. 1989, ISBN 3-7643-2229-2 .
• Herbert Meschkowski : The way of thinking of great mathematicians. A path to the history of mathematics . Vieweg, Braunschweig 1990, ISBN 3-528-28179-0 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 .
• Tammo tom Dieck : Topology . 2nd, completely revised and expanded edition. de Gruyter, Berlin et al. 2000, ISBN 3-11-016236-9 .

Web link

• Link to the original work by Harzheim on " Generalization of the decomposition theorem from Jordan-Brouwer-Alexander to products of linearly ordered continuums "

Individual evidence

1. ^ KH Mayer: Algebraic Topology . 1989, p. 254 .
2. ^ E. Harzheim: Introduction to the combinatorial topology . 1978, p. 141 ff .
3. ^ Proof of Jordan-Brouwer Separation Theorem math.berkeley.edu, November 20, 2014, accessed on September 7, 2019
4. ↑ Jordan-Brouwer's decomposition theorem according to ”Differential Topology”, V. Guillemin / A. Pollack , bell0bytes.eu, January 14, 2007, accessed on September 7, 2019
5. ^ E. Harzheim: Introduction to the combinatorial topology . 1978, p. 153 .
6. ^ H. Meschkowski: Thoughts of great mathematicians . 1990, p. 246 .
7. ^ JW Alexander: A proof and extension of the Jordan-Brouwer separation theorem . 1922, p. 333 ff .
8. ^ H. Schubert: Topology . 1975, p. 272 .
9. ↑ E. Sperner: New proof for the invariance of the dimensional number and the area . 1928, p. 265 ff .