Banach's illustration theorem

The Banach mapping theorem is a mathematical theorem from the field of set theory named after the Polish mathematician Stefan Banach . The sentence deals with a fundamental property of images . It is closely linked to the Cantor-Bernstein-Schröder theorem .

Formulation of the sentence

The sentence can be formulated as follows:

Given are sets     and     and corresponding figures ${\displaystyle M}$ ${\displaystyle N}$

  ${\displaystyle {\phi }\colon \,M\to N}$   and   . ${\displaystyle {\psi }\colon \,N\to M}$

Let be   injective . ${\displaystyle {\phi }}$  

Then sets exist   ${\displaystyle M_{1},M_{2},N_{1},N_{2}}$   with

  ${\displaystyle M=M_{1}{\cup }M_{2}}$  and   ${\displaystyle M_{1}{\cap }M_{2}=\emptyset }$ 

such as

  ${\displaystyle N=N_{1}{\cup }N_{2}}$  and   ${\displaystyle N_{1}{\cap }N_{2}=\emptyset }$ 

in such a way that:

  ${\displaystyle {\phi }(M_{1})=N_{1}}$  and   ${\displaystyle {\psi }(N_{2})=M_{2}}$ 

Tightening

With the help of the fixed point theorem of Tarski and Knaster it can be shown that the claim of the theorem still holds if the injectivity condition for the mapping     is dropped. ${\displaystyle {\phi }\colon \,M\to N}$

The Banach figure set (strengthened version) would read as follows:

Given are sets     and     and corresponding figures ${\displaystyle M}$ ${\displaystyle N}$

  ${\displaystyle {\phi }\colon \,M\to N}$   and    . ${\displaystyle {\psi }\colon \,N\to M}$

Then sets exist   ${\displaystyle M_{1},M_{2},N_{1},N_{2}}$   with

  ${\displaystyle M=M_{1}{\cup }M_{2}}$  and   ${\displaystyle M_{1}{\cap }M_{2}=\emptyset }$ 

such as

  ${\displaystyle N=N_{1}{\cup }N_{2}}$  and   ${\displaystyle N_{1}{\cap }N_{2}=\emptyset }$ 

in such a way that:

  ${\displaystyle {\phi }(M_{1})=N_{1}}$  and   ${\displaystyle {\psi }(N_{2})=M_{2}}$ 

Proof (tightening)

Look at the picture with . ${\displaystyle F:{\mathcal {P}}(M)\rightarrow {\mathcal {P}}(M)}$ ${\displaystyle F(A):=M\setminus (\psi (N\setminus \phi (A)))}$

Since is monotonic, has a fixed point according to the fixed point theorem of Tarski and Knaster . So it applies or equivalently ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle M_{1}}$ ${\displaystyle M_{1}=M\setminus (\psi (N\setminus \phi (M_{1})))}$

${\displaystyle M\setminus M_{1}=\psi (N\setminus \phi (M_{1}))}$.

We now set , and . ${\displaystyle M_{2}:=M\setminus M_{1}}$ ${\displaystyle N_{1}:=\phi (M_{1})}$ ${\displaystyle N_{2}:=N\setminus N_{1}}$

We hereby receive as requested and . ${\displaystyle {\phi }(M_{1})=N_{1}}$ ${\displaystyle {\psi }(N_{2})=\psi (N\setminus \phi (M_{1}))=M\setminus M_{1}=M_{2}}$

Inference

The Cantor-Bernstein-Schröder theorem follows directly from Banach's mapping theorem .

literature

Articles and original works

• Stefan Banach : Un théorème sur les transformations biunivoques . In: Fundamenta Mathematicae . 6, 1924, pp. 236-239.
• Alfred Tarski : A lattice-theoretical fixpoint theorem and its applications . In: Pacific Journal of Mathematics . 5, 1955, pp. 285-309.
• Bronislaw Knaster : Un théorème sur les fonctions d'ensembles . In: Ann. Soc. Polon. Math. . 6, 1928, pp. 133-134.

Monographs

• Garrett Birkhoff : Lattice Theory . 3. Edition. American Mathematical Society, Providence, Rhode Island 1979.
• Heinz Lüneburg : Combinatorics . Birkhäuser Verlag, Basel u. a. 1971, ISBN 3-7643-0548-7 .
• Heinz Lüneburg : Tools and Fundamental Constructions of Combinatorial Mathematics . BI Wissenschaftsverlag, Mannheim u. a. 1989, ISBN 3-411-03194-8 .

Individual evidence

1. ^ Stefan Banach : Un théorème sur les transformations biunivoques . In: Fundamenta Mathematicae . tape 6 , 1924, pp. 236-239 .
2. ^ Heinz Lüneburg : Combinatorics . Birkhäuser Verlag, Basel u. a. 1971, ISBN 3-7643-0548-7 , pp. 65 .
3. ^ Heinz Lüneburg : Tools and Fundamental Constructions of Combinatorial Mathematics . BI Wissenschaftsverlag, Mannheim u. a. 1989, ISBN 3-411-03194-8 , pp. 348-349 .
4. ^ Stefan Banach : Un théorème sur les transformations biunivoques . In: Fundamenta Mathematicae . tape 6 , 1924, introduction, p. 236 .
5. ^ Heinz Lüneburg : Combinatorics . Birkhäuser Verlag, Basel u. a. 1971, ISBN 3-7643-0548-7 , pp. 66 .
6. ^ Heinz Lüneburg : Tools and Fundamental Constructions of Combinatorial Mathematics . BI Wissenschaftsverlag, Mannheim u. a. 1989, ISBN 3-411-03194-8 , pp. 349 .