1) Si
M
=
R
n
{\displaystyle M=\mathbb {R} ^{n}}
(
R
n
,
i
d
:
R
n
→
R
n
)
{\displaystyle (\mathbb {R} ^{n},\;id:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n})}
∀
n
∈
N
:
n
>
0
{\displaystyle \forall n\in \mathbb {N} :n>0}
2) Si
M
=
R
{\displaystyle M=\mathbb {R} }
(
(
a
,
b
)
,
i
:
(
a
,
b
)
↪
R
)
{\displaystyle ((a,b),\;i:(a,b)\hookrightarrow \mathbb {R} )}
∀
a
,
b
∈
R
:
a
<
b
{\displaystyle \forall a,b\in \mathbb {R} :a<b}
3) Si
M
=
R
{\displaystyle M=\mathbb {R} }
(
R
,
x
↦
x
3
)
{\displaystyle (\mathbb {R} ,\;x\mapsto x^{3})}
x
2
n
+
1
∀
n
>
1
{\displaystyle x^{2n+1}\forall n>1}
Demostración:
R
{\displaystyle \mathbb {R} }
∀
x
∈
R
,
∃
!
x
3
,
∃
!
x
3
{\displaystyle \forall x\in \mathbb {R} ,\;\exists !x^{3},\exists !{\sqrt[{3}]{x}}}
4) Si
M
=
S
1
⊂
R
2
≅
C
{\displaystyle M=S^{1}\subset \mathbb {R^{2}} \cong \mathbb {C} }
(
S
1
{
i
}
,
ϕ
)
{\displaystyle (S^{1}\ \{i\},\;\phi )}
ϕ
:
S
1
{
i
}
⟶
(
−
3
π
2
,
π
2
)
z
⟼
θ
:=
det
−
arg
(
−
3
π
2
,
π
2
)
(
z
)
{\displaystyle {\begin{matrix}\phi :&{S^{1}\ \{i\}}&\longrightarrow {}&{(-{\frac {3\pi }{2}},{\frac {\pi }{2}})}\\&z&\longmapsto &{\theta :=\det -\arg _{(-{\frac {3\pi }{2}},{\frac {\pi }{2}})}(z)}\end{matrix}}}
5) Si
M
=
S
1
⊂
R
2
≅
C
{\displaystyle M=S^{1}\subset \mathbb {R^{2}} \cong \mathbb {C} }
(
S
1
{
i
}
,
ψ
)
{\displaystyle (S^{1}\ \{i\},\;\psi )}
la proyección estereográfica
ψ
:
S
1
{
i
}
⟶
R
z
⟼
x
:=
c
o
s
(
a
r
g
(
z
)
)
1
−
s
e
n
(
a
r
g
(
z
)
)
{\displaystyle {\begin{matrix}\psi :&{S^{1}\ \{i\}}&\longrightarrow {}&\mathbb {R} \\&z&\longmapsto &{x:={\frac {cos(arg(z))}{1-sen(arg(z))}}}\end{matrix}}}
Caso particular en el que n=2 6) Si
M
=
S
n
⊂
R
n
+
1
{\displaystyle M=S^{n}\subset \mathbb {R} ^{n+1}}
(
S
n
{
(
0
,
…
,
0
,
1
)
}
,
ϕ
)
{\displaystyle (S^{n}\ \{(0,\;\dots ,\;0,\;1)\},\;\phi )}
ϕ
:
S
n
{
(
0
,
…
,
0
,
1
)
}
⟶
R
n
(
x
1
,
…
,
x
n
+
1
)
⟼
(
x
1
,
…
,
x
n
)
1
−
x
n
+
1
{\displaystyle {\begin{matrix}\phi :&{S^{n}\ \{(0,\;\dots ,\;0,\;1)\}}&\longrightarrow {}&\mathbb {R} ^{n}\\&(x_{1},\;\dots ,\;x_{n+1})&\longmapsto &{\frac {(x_{1},\;\dots ,\;x_{n})}{1-x_{n+1}}}\end{matrix}}}
![{\displaystyle \forall x\in \mathbb {R} ,\;\exists !x^{3},\exists !{\sqrt[{3}]{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db2b44a190505392b05080098b4fa74ba7141301)
