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 .
R.
{\ displaystyle \ mathbb {R}}
I.
1
,
I.
2
,
I.
3
{\ displaystyle I_ {1} \;, \; I_ {2} \;, \; I_ {3}}
It is assumed that
⋂
k
∈
{
1
,
2
,
3
}
I.
k
∘
≠
∅
{\ 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:
∃
j
∈
{
1
,
2
,
3
}
:
(
I.
j
∘
⊆
⋃
k
∈
{
1
,
2
,
3
}
k
≠
j
I.
k
∘
)
{\ 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 .
I.
k
=
[
a
k
,
b
k
]
{\ displaystyle I_ {k} = \ left [a_ {k}, b_ {k} \ right]}
k
=
1
,
2
,
3
{\ displaystyle k = 1,2,3}
I.
k
∘
=
(
a
k
,
b
k
)
{\ displaystyle I_ {k} ^ {\ circ} = \ left (a_ {k}, b_ {k} \ right)}
Be so and be so . Then applies to .
i
∈
{
1
,
2
,
3
}
{\ displaystyle i \ in \ left \ {1,2,3 \ right \}}
a
i
=
min
(
a
1
,
a
2
,
a
3
)
{\ displaystyle a_ {i} = \ min (a_ {1}, a_ {2}, a_ {3})}
j
∈
{
1
,
2
,
3
}
{\ displaystyle j \ in \ left \ {1,2,3 \ right \}}
b
j
=
Max
(
b
1
,
b
2
,
b
3
)
{\ displaystyle b_ {j} = \ max (b_ {1}, b_ {2}, b_ {3})}
I.
k
⊂
[
a
i
,
b
j
]
{\ displaystyle I_ {k} \ subset \ left [a_ {i}, b_ {j} \ right]}
k
=
1
,
2
,
3
{\ displaystyle k = 1,2,3}
Case 1: . Then applies and for .
i
=
j
{\ displaystyle i = j}
I.
k
⊂
I.
i
{\ displaystyle I_ {k} \ subset I_ {i}}
I.
k
∘
⊂
I.
i
∘
{\ displaystyle I_ {k} ^ {\ circ} \ subset I_ {i} ^ {\ circ}}
k
=
1
,
2
,
3
{\ 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
i
≠
j
{\ displaystyle i \ not = j}
z
{\ displaystyle z}
z
∈
(
a
i
,
b
i
)
{\ displaystyle z \ in \ left (a_ {i}, b_ {i} \ right)}
z
∈
(
a
j
,
b
j
)
{\ displaystyle z \ in \ left (a_ {j}, b_ {j} \ right)}
a
j
<
z
<
b
i
{\ displaystyle a_ {j} <z <b_ {i}}
I.
k
⊂
[
a
i
,
b
j
]
=
[
a
i
,
z
]
∪
[
z
,
b
j
]
⊂
[
a
i
,
b
i
]
∪
[
a
j
,
b
j
]
=
I.
i
∪
I.
j
{\ 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 .
k
=
1
,
2
,
3
{\ displaystyle k = 1,2,3}
x
∈
I.
k
∘
{\ displaystyle x \ in I_ {k} ^ {\ circ}}
a
k
<
x
<
b
k
{\ displaystyle a_ {k} <x <b_ {k}}
x
≤
z
{\ displaystyle x \ leq z}
x
>
z
{\ displaystyle x> z}
x
≤
z
{\ displaystyle x \ leq z}
a
i
≤
a
k
<
x
≤
z
<
b
i
{\ displaystyle a_ {i} \ leq a_ {k} <x \ leq z <b_ {i}}
x
∈
I.
i
∘
{\ displaystyle x \ in I_ {i} ^ {\ circ}}
x
>
z
{\ displaystyle x> z}
a
j
<
z
<
x
<
b
j
{\ displaystyle a_ {j} <z <x <b_ {j}}
x
∈
I.
j
∘
{\ displaystyle x \ in I_ {j} ^ {\ circ}}
literature
References and comments
↑ 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
^ Wacław Sierpiński: Cardinal and Ordinal Numbers (= monograph matematyczne . Volume 34 ). 2nd Edition. PWN - Polish Scientific Publishers, Warsaw 1965, p. 41 ( MR0194339 ).
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