In mathematics is composite (ger .: join ) topological spaces one on John Milnor declining construction of the topology .
construction
The combination of two
intervals (blue and green) is a 3-dimensional polytope (gray).
Association of two topological spaces
Let and be two topological spaces. Their network is defined as follows. The elements of are the couple
X
1
{\ displaystyle X_ {1}}
X
2
{\ displaystyle X_ {2}}
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})}
with ,
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}
where is an abbreviation for the couple and for all and all
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})}
and
(
1
x
1
,
0
x
2
)
=
(
1
x
1
,
0
x
2
′
)
{\ displaystyle (1x_ {1}, 0x_ {2}) = (1x_ {1}, 0x_ {2} ^ {\ prime})}
is set. (So all points from are clearly connected to all points from by stretching the length .)
X
1
{\ displaystyle X_ {1}}
X
2
{\ displaystyle X_ {2}}
1
{\ displaystyle 1}
The topology on is by definition the coarsest topology (the topology with the fewest open sets) with respect to all coordinate mappings
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)}
and
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)}
are steady.
Examples
The compound of a room with a point is the cone above .
X
{\ displaystyle X}
C.
X
{\ displaystyle CX}
X
{\ displaystyle X}
The connection of a room with the 2-element room is the suspension of .
X
{\ displaystyle X}
S.
0
{\ displaystyle S ^ {0}}
S.
X
{\ displaystyle SX}
X
{\ displaystyle X}
The union of two spheres and is the -dimensional sphere .
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}}
The compound of circles is the -dimensional sphere .
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}}
For the Cartesian product of two CAT (0) -spaces and their geodetic boundaries holds .
X
1
×
X
2
{\ displaystyle X_ {1} \ times X_ {2}}
X
1
,
X
2
{\ displaystyle X_ {1}, X_ {2}}
∂
∞
(
X
1
×
X
2
)
=
∂
∞
X
1
∗
∂
∞
X
2
{\ displaystyle \ partial _ {\ infty} (X_ {1} \ times X_ {2}) = \ partial _ {\ infty} X_ {1} * \ partial _ {\ infty} X_ {2}}
Spherical compound
On the connection of two metric spaces and one can define a metric as follows: The distance is the number in the interval for which
(
X
1
,
d
1
)
{\ displaystyle (X_ {1}, d_ {1})}
(
X
2
,
d
2
)
{\ displaystyle (X_ {2}, d_ {2})}
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 \})}
applies. Note that the constraints on this metric are based on and not the original metrics , but rather give.
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 \}}
The metric space is called the spherical compound of the metric spaces and .
(
X
1
∗
X
2
,
d
)
{\ displaystyle (X_ {1} * X_ {2}, d)}
(
X
1
,
d
1
)
{\ displaystyle (X_ {1}, d_ {1})}
(
X
2
,
d
2
)
{\ displaystyle (X_ {2}, d_ {2})}
Association of an infinite number of topological spaces
Let it be a family of topological spaces. The elements of the compound are the tuples
{
X
j
:
j
∈
J
}
{\ displaystyle \ left \ {X_ {j} \ colon j \ in J \ right \}}
X
=
∗
j
∈
J
X
j
{\ displaystyle X = * _ {y \ in J} X_ {j}}
J
{\ displaystyle J}
(
t
j
x
j
:
j
∈
J
)
{\ displaystyle (t_ {j} x_ {j} \ colon j \ in J)}
with almost everyone .
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,}
t
j
=
0
{\ displaystyle t_ {j} = 0}
Two tuples and define the same element if and only if:
(
t
j
x
j
)
{\ displaystyle (t_ {j} x_ {j})}
(
u
j
y
j
)
{\ displaystyle (u_ {j} y_ {j})}
For everyone is .
j
∈
J
{\ displaystyle j \ in J}
t
j
=
u
j
{\ displaystyle t_ {j} = u_ {j}}
For all true: .
j
∈
J
{\ displaystyle j \ in J}
t
j
≠
0
⟹
x
j
=
y
j
{\ displaystyle t_ {j} \ not = 0 \ Longrightarrow x_ {j} = y_ {j}}
The topology on is the coarsest topology (the topology with the fewest open sets) with respect to all coordinate mappings
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)}
and
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)}
are steady.
Examples
literature
Tammo tom Dieck: Topology. de Gruyter textbook. 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. Basic teaching of the mathematical sciences, 319. Springer, Berlin 1999, ISBN 3-540-64324-9
Individual evidence
↑ Berestovskiĭ, VN: Borsuk's problem on metrization of a polyhedron. (Russian) Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 273-277.
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