Intervallum est copia numerorum realium quae, pro quibuscumque numeris in ea copia, omnes numeros inter illos numeros continet. Ergo
I
{\displaystyle I}
a
,
b
∈
I
,
a
≤
x
≤
b
⇒
x
∈
I
{\displaystyle a,b\in I,a\leq x\leq b\Rightarrow x\in I}
Genera intervallorum
recensere
Tria genera intervallorum sunt:
Clausum:
[
a
,
b
]
:=
{
x
∈
R
:
a
≤
x
≤
b
}
.
{\displaystyle [a,b]:=\lbrace x\in \mathbb {R} :a\leq x\leq b\rbrace .}
Apertum:
]
a
,
b
[
:=
(
a
,
b
)
:=
{
x
∈
R
:
a
<
x
<
b
}
.
{\displaystyle ]a,b[\ :=(a,b):=\lbrace x\in \mathbb {R} :a<x<b\rbrace .}
Semi-apertum vel semi-clausum:
[
a
,
b
[
:=
[
a
,
b
)
:=
{
x
∈
R
:
a
≤
x
<
b
}
{\displaystyle [a,b[\ :=[a,b):=\lbrace x\in \mathbb {R} :a\leq x<b\rbrace }
]
a
,
b
]
:=
(
a
,
b
]
:=
{
x
∈
R
:
a
<
x
≤
b
}
.
{\displaystyle ]a,b]:=(a,b]:=\lbrace x\in \mathbb {R} :a<x\leq b\rbrace .}
Finis intervalli aperti etiam infinitas esse potest:
[
a
,
∞
[
:=
[
a
,
∞
)
:=
{
x
∈
R
:
x
≥
a
}
{\displaystyle [a,\infty [\ :=[a,\infty ):=\lbrace x\in \mathbb {R} :x\geq a\rbrace }
]
a
,
∞
[
:=
(
a
,
∞
)
:=
{
x
∈
R
:
x
>
a
}
{\displaystyle ]a,\infty [\ :=(a,\infty ):=\lbrace x\in \mathbb {R} :x>a\rbrace }
]
∞
,
b
]
:=
(
∞
,
b
]
:=
{
x
∈
R
:
x
≤
b
}
{\displaystyle ]\infty ,b]:=(\infty ,b]:=\lbrace x\in \mathbb {R} :x\leq b\rbrace }
]
∞
,
b
[
:=
(
∞
,
b
)
:=
{
x
∈
R
:
x
<
b
}
{\displaystyle ]\infty ,b[:=(\infty ,b):=\lbrace x\in \mathbb {R} :x<b\rbrace }
]
∞
,
∞
[
:=
(
∞
,
∞
)
:=
R
.
{\displaystyle ]\infty ,\infty [\ :=(\infty ,\infty ):=\mathbb {R} .}
Casus singulares sunt:
Si
b
>
a
{\displaystyle b>a}
vacua :
[
b
,
a
]
=
(
a
,
a
)
=
[
a
,
a
)
=
(
a
,
a
]
=
{
}
=
∅
{\displaystyle [b,a]=(a,a)=[a,a)=(a,a]=\lbrace \rbrace =\emptyset }
[
a
,
a
]
=
{
a
}
.
{\displaystyle [a,a]=\lbrace a\rbrace .}
Nexus interni
Interval and segment. Encyclopedia of Mathematics. [1] 
![{\displaystyle [a,b]:=\lbrace x\in \mathbb {R} :a\leq x\leq b\rbrace .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b618090d9cd92f54e7a537ecef8f127fd6c8b10a)
![{\displaystyle ]a,b[\ :=(a,b):=\lbrace x\in \mathbb {R} :a<x<b\rbrace .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea05be6abc1aa851b451d415a5390b686ed72efb)
![{\displaystyle ]a,b]:=(a,b]:=\lbrace x\in \mathbb {R} :a<x\leq b\rbrace .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b30baf30678e117c2d433c511027d88cf42a6fb)
![{\displaystyle ]a,\infty [\ :=(a,\infty ):=\lbrace x\in \mathbb {R} :x>a\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af830721f1ca2e3ce1d6b5e8030d423bc9d386b)
![{\displaystyle ]\infty ,b]:=(\infty ,b]:=\lbrace x\in \mathbb {R} :x\leq b\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e261f0d075a89333ba267d4021d9f179c2e94b2)
![{\displaystyle ]\infty ,b[:=(\infty ,b):=\lbrace x\in \mathbb {R} :x<b\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/451601c2d906b86c220eafae571f04e6791a289d)
![{\displaystyle ]\infty ,\infty [\ :=(\infty ,\infty ):=\mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e8ed7fff63b397796244f8e1386e7b27c516dc5)
![{\displaystyle [b,a]=(a,a)=[a,a)=(a,a]=\lbrace \rbrace =\emptyset }](https://wikimedia.org/api/rest_v1/media/math/render/svg/da7fd5c46895431d63d6b3066ca14eb2bb53b3b4)
![{\displaystyle [a,a]=\lbrace a\rbrace .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc9f7144fcc5fb67fb4f6af10a397089e766b6b)