Denjoy's theorem (topology)

The set of Denjoy is a mathematical theorem that the French mathematician Arnaud Denjoy back. It deals with a fundamental connected property of the topology of real numbers . Denjoy published it in 1915.

formulation

The sentence can be stated as follows:

Given are     three compact intervals   . ${\displaystyle \mathbb {R} }$   ${\displaystyle I_{1}\;,\;I_{2}\;,\;I_{3}}$

It is assumed that

${\displaystyle \bigcap \limits _{k\;\in \;\{1\;,\;2\;,\;3\} \atop \;}{I_{k}^{\circ }}\;\neq \;\emptyset }$

assume that the three intervals have at least one internal point in common .

Then:

There are among the three intervals at least one of which is so from the other two intervals covers , is that each of its interior points at the same time an interior point is one of the other two intervals.

In formulas:

${\displaystyle \exists j\in \{1\;,\;2\;,\;3\}\colon {\bigl (}I_{j}^{\circ }\;\subseteq \;\bigcup \limits _{k\;\in \;\{1\;,\;2\;,\;3\} \atop \;\ k\;\neq \;j}{I_{k}^{\circ }}{\bigr )}}$  

proof

The compact intervals are of the form for . Your inside is each . ${\displaystyle I_{k}=\left[a_{k},b_{k}\right]}$ ${\displaystyle k=1,2,3}$ ${\displaystyle I_{k}^{\circ }=\left(a_{k},b_{k}\right)}$

Be so and be so . Then applies to . ${\displaystyle i\in \left\{1,2,3\right\}}$ ${\displaystyle a_{i}=\min(a_{1},a_{2},a_{3})}$ ${\displaystyle j\in \left\{1,2,3\right\}}$ ${\displaystyle b_{j}=\max(b_{1},b_{2},b_{3})}$ ${\displaystyle I_{k}\subset \left[a_{i},b_{j}\right]}$ ${\displaystyle k=1,2,3}$

Case 1: . Then applies and for . ${\displaystyle i=j}$ ${\displaystyle I_{k}\subset I_{i}}$ ${\displaystyle I_{k}^{\circ }\subset I_{i}^{\circ }}$ ${\displaystyle k=1,2,3}$

Case 2: . According to the assumption there is a common point in the interior of the three intervals, for which in particular and , therefore, applies. It follows ${\displaystyle i\not =j}$ ${\displaystyle z}$ ${\displaystyle z\in \left(a_{i},b_{i}\right)}$ ${\displaystyle z\in \left(a_{j},b_{j}\right)}$ ${\displaystyle a_{j}<z<b_{i}}$

${\displaystyle I_{k}\subset \left[a_{i},b_{j}\right]=\left[a_{i},z\right]\cup \left[z,b_{j}\right]\subset \left[a_{i},b_{i}\right]\cup \left[a_{j},b_{j}\right]=I_{i}\cup I_{j}}$

for . For an inner point one has and either or . In the case follows and with it , in the case follows and with it . ${\displaystyle k=1,2,3}$ ${\displaystyle x\in I_{k}^{\circ }}$ ${\displaystyle a_{k}<x<b_{k}}$ ${\displaystyle x\leq z}$ ${\displaystyle x>z}$ ${\displaystyle x\leq z}$ ${\displaystyle a_{i}\leq a_{k}<x\leq z<b_{i}}$ ${\displaystyle x\in I_{i}^{\circ }}$ ${\displaystyle x>z}$ ${\displaystyle a_{j}<z<x<b_{j}}$ ${\displaystyle x\in I_{j}^{\circ }}$

literature

• Wacław Sierpiński: Cardinal and Ordinal Numbers (=  monograph matematyczne . Volume 34 ). 2nd Edition. PWN - Polish Scientific Publishers, Warsaw 1965 ( MR0194339 ).
• Egbert Harzheim : Ordered Sets (=  Advances in Mathematics . Volume 7 ). Springer Verlag, New York 2005, ISBN 0-387-24219-8 ( MR2127991 ).
• Horst Schubert : Topology. An introduction (=  mathematical guidelines ). 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

References and comments

1. ↑ Denjoy: Mémoire sur les nombres dérivés des fonctions continues . In: Journal de Mathématiques Pures et Appliquées , Series 7, Volume 1, 1915, pp. 105–240, mathdoc.fr , here p. 223
2. ^ Wacław Sierpiński: Cardinal and Ordinal Numbers (=  monograph matematyczne . Volume 34 ). 2nd Edition. PWN - Polish Scientific Publishers, Warsaw 1965, p. 41 ( MR0194339 ).