In der Mathematik ist der Verbund (engl.: join ) topologischer Räume eine auf John Milnor zurückgehende Konstruktion aus der Topologie .
Der Verbund zweier Intervalle (blau und grün) ist ein 3-dimensionales Polytop (grau). Es seien
X
1
{\displaystyle X_{1}}
X
2
{\displaystyle X_{2}}
Verbund
X
=
X
1
∗
X
2
{\displaystyle X=X_{1}*X_{2}}
X
{\displaystyle X}
(
t
1
x
1
,
t
2
x
2
)
{\displaystyle (t_{1}x_{1},t_{2}x_{2})}
x
1
∈
X
1
,
x
2
∈
X
2
,
t
1
,
t
2
∈
[
0
,
1
]
,
t
1
+
t
2
=
1
{\displaystyle x_{1}\in X_{1},x_{2}\in X_{2},t_{1},t_{2}\in \left[0,1\right],t_{1}+t_{2}=1}
wobei
t
i
x
i
{\displaystyle t_{i}x_{i}}
(
t
i
,
x
i
)
{\displaystyle (t_{i},x_{i})}
x
1
,
x
1
′
∈
X
1
{\displaystyle x_{1},x_{1}^{\prime }\in X_{1}}
x
2
,
x
2
′
∈
X
2
{\displaystyle x_{2},x_{2}^{\prime }\in X_{2}}
(
0
x
1
,
1
x
2
)
=
(
0
x
1
′
,
1
x
2
)
{\displaystyle (0x_{1},1x_{2})=(0x_{1}^{\prime },1x_{2})}
(
1
x
1
,
0
x
2
)
=
(
1
x
1
,
0
x
2
′
)
{\displaystyle (1x_{1},0x_{2})=(1x_{1},0x_{2}^{\prime })}
gesetzt wird. (Anschaulich werden also alle Punkte aus
X
1
{\displaystyle X_{1}}
X
2
{\displaystyle X_{2}}
1
{\displaystyle 1}
Die Topologie auf
X
{\displaystyle X}
t
i
:
X
→
[
0
,
1
]
{\displaystyle t_{i}\colon X\to \left[0,1\right]}
(
t
1
x
1
,
t
2
x
2
)
→
t
i
(
i
=
1
,
2
)
{\displaystyle (t_{1}x_{1},t_{2}x_{2})\to t_{i}\quad (i=1,2)}
und
x
i
:
{
(
t
1
x
1
,
t
2
x
2
)
:
t
i
≠
0
}
→
X
i
{\displaystyle x_{i}\colon \left\{(t_{1}x_{1},t_{2}x_{2})\colon t_{i}\not =0\right\}\to X_{i}}
(
t
1
x
1
,
t
2
x
2
)
→
x
i
(
i
=
1
,
2
)
{\displaystyle (t_{1}x_{1},t_{2}x_{2})\to x_{i}\quad (i=1,2)}
stetig sind.
Der Verbund eines Raumes
X
{\displaystyle X}
Kegel
C
X
{\displaystyle CX}
X
{\displaystyle X}
Der Verbund eines Raumes
X
{\displaystyle X}
S
0
{\displaystyle S^{0}}
Einhängung
S
X
{\displaystyle SX}
X
{\displaystyle X}
Der Verbund zweier Sphären
S
k
{\displaystyle S^{k}}
S
l
{\displaystyle S^{l}}
(
k
+
l
+
1
)
{\displaystyle (k+l+1)}
S
k
+
l
+
1
{\displaystyle S^{k+l+1}}
Der Verbund von
k
{\displaystyle k}
S
1
∗
…
∗
S
1
{\displaystyle S^{1}*\ldots *S^{1}}
(
2
k
−
1
)
{\displaystyle (2k-1)}
S
2
k
−
1
{\displaystyle S^{2k-1}}
Für das kartesische Produkt
X
1
×
X
2
{\displaystyle X_{1}\times X_{2}}
CAT(0)-Räume
X
1
,
X
2
{\displaystyle X_{1},X_{2}}
geodätische Ränder gilt
∂
∞
(
X
1
×
X
2
)
=
∂
∞
X
1
∗
∂
∞
X
2
{\displaystyle \partial _{\infty }(X_{1}\times X_{2})=\partial _{\infty }X_{1}*\partial _{\infty }X_{2}}
Auf dem Verbund zweier metrischer Räume
(
X
1
,
d
1
)
{\displaystyle (X_{1},d_{1})}
(
X
2
,
d
2
)
{\displaystyle (X_{2},d_{2})}
[ 1]
d
(
(
t
1
x
1
,
t
2
x
2
)
,
(
s
1
y
1
,
s
2
y
2
)
)
{\displaystyle d((t_{1}x_{1},t_{2}x_{2}),(s_{1}y_{1},s_{2}y_{2}))}
[
0
,
π
]
{\displaystyle \left[0,\pi \right]}
cos
(
d
(
(
t
1
x
1
,
t
2
x
2
)
,
(
s
1
y
1
,
s
2
y
2
)
)
)
=
t
1
s
1
cos
(
m
i
n
{
π
,
d
1
(
x
1
,
y
1
)
}
)
+
t
2
s
2
cos
(
m
i
n
{
π
,
d
2
(
x
2
,
y
2
)
}
)
{\displaystyle \cos(d((t_{1}x_{1},t_{2}x_{2}),(s_{1}y_{1},s_{2}y_{2})))={\sqrt {t_{1}s_{1}}}\cos(min\left\{\pi ,d_{1}(x_{1},y_{1})\right\})+{\sqrt {t_{2}s_{2}}}\cos(min\left\{\pi ,d_{2}(x_{2},y_{2})\right\})}
gilt. Man beachte, dass die Einschränkungen dieser Metrik auf
X
1
{\displaystyle X_{1}}
X
2
{\displaystyle X_{2}}
d
i
(
x
,
y
)
{\displaystyle d_{i}(x,y)}
m
i
n
{
π
,
d
i
(
x
,
y
)
}
{\displaystyle min\left\{\pi ,d_{i}(x,y)\right\}}
Der metrische Raum
(
X
1
∗
X
2
,
d
)
{\displaystyle (X_{1}*X_{2},d)}
sphärischer Verbund der metrischen Räume
(
X
1
,
d
1
)
{\displaystyle (X_{1},d_{1})}
(
X
2
,
d
2
)
{\displaystyle (X_{2},d_{2})}
Es sei
{
X
j
:
j
∈
J
}
{\displaystyle \left\{X_{j}\colon j\in J\right\}}
X
=
∗
j
∈
J
X
j
{\displaystyle X=*_{j\in J}X_{j}}
J
{\displaystyle J}
(
t
j
x
j
:
j
∈
J
)
{\displaystyle (t_{j}x_{j}\colon j\in J)}
t
j
∈
[
0
,
1
]
,
x
j
∈
X
j
,
∑
j
∈
J
t
j
=
1
,
{\displaystyle t_{j}\in \left[0,1\right],x_{j}\in X_{j},\sum _{j\in J}t_{j}=1,}
fast alle
t
j
=
0
{\displaystyle t_{j}=0}
Zwei Tupel
(
t
j
x
j
)
{\displaystyle (t_{j}x_{j})}
(
u
j
y
j
)
{\displaystyle (u_{j}y_{j})}
Für alle
j
∈
J
{\displaystyle j\in J}
t
j
=
u
j
{\displaystyle t_{j}=u_{j}}
Für alle
j
∈
J
{\displaystyle j\in J}
t
j
≠
0
⟹
x
j
=
y
j
{\displaystyle t_{j}\not =0\Longrightarrow x_{j}=y_{j}}
Die Topologie auf
X
{\displaystyle X}
t
j
:
X
→
[
0
,
1
]
{\displaystyle t_{j}\colon X\to \left[0,1\right]}
(
t
i
x
i
)
→
t
j
(
j
∈
J
)
{\displaystyle (t_{i}x_{i})\to t_{j}\quad (j\in J)}
und
x
j
:
{
(
t
i
x
i
)
:
t
j
≠
0
}
→
X
j
{\displaystyle x_{j}\colon \left\{(t_{i}x_{i})\colon t_{j}\not =0\right\}\to X_{j}}
(
t
i
x
i
)
→
x
j
(
j
∈
J
)
{\displaystyle (t_{i}x_{i})\to x_{j}\quad (j\in J)}
stetig sind.
Tammo tom Dieck: Topologie. de Gruyter Lehrbuch. Walter de Gruyter & Co., Berlin 1991, ISBN 3-11-013187-0 ; 3-11-012463-7
Martin R. Bridson; André Haefliger: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer, Berlin 1999, ISBN 3-540-64324-9
↑ Berestovskiĭ, V. N.: Borsuk's problem on metrization of a polyhedron. (russisch) Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 273–277.
![{\displaystyle x_{1}\in X_{1},x_{2}\in X_{2},t_{1},t_{2}\in \left[0,1\right],t_{1}+t_{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/503095e06b2873ac20fbc392eb64ba6c3f799008)
![{\displaystyle t_{i}\colon X\to \left[0,1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b765295ca2f0766dfbc895c61cb10def78ba084)
![{\displaystyle \left[0,\pi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e34615383b8ffe6f079cfd094c2b274fab51a4)
![{\displaystyle t_{j}\in \left[0,1\right],x_{j}\in X_{j},\sum _{j\in J}t_{j}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf32d88a876263fc782f07274b913c324f6b492a)
![{\displaystyle t_{j}\colon X\to \left[0,1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c14f303ba2519d040411eaea1d84103c1c66fb68)
