- Source: Restricted product
In mathematics, the restricted product is a construction in the theory of topological groups.
Let
I
{\displaystyle I}
be an index set;
S
{\displaystyle S}
a finite subset of
I
{\displaystyle I}
. If
G
i
{\displaystyle G_{i}}
is a locally compact group for each
i
∈
I
{\displaystyle i\in I}
, and
K
i
⊂
G
i
{\displaystyle K_{i}\subset G_{i}}
is an open compact subgroup for each
i
∈
I
∖
S
{\displaystyle i\in I\setminus S}
, then the restricted product
∏
i
′
G
i
{\displaystyle \prod _{i}\nolimits 'G_{i}\,}
is the subset of the product of the
G
i
{\displaystyle G_{i}}
's consisting of all elements
(
g
i
)
i
∈
I
{\displaystyle (g_{i})_{i\in I}}
such that
g
i
∈
K
i
{\displaystyle g_{i}\in K_{i}}
for all but finitely many
i
∈
I
∖
S
{\displaystyle i\in I\setminus S}
.
This group is given the topology whose basis of open sets are those of the form
∏
i
A
i
,
{\displaystyle \prod _{i}A_{i}\,,}
where
A
i
{\displaystyle A_{i}}
is open in
G
i
{\displaystyle G_{i}}
and
A
i
=
K
i
{\displaystyle A_{i}=K_{i}}
for all but finitely many
i
{\displaystyle i}
.
One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field.
See also
Direct sum
References
Fröhlich, A.; Cassels, J. W. (1967), Algebraic number theory, Boston, MA: Academic Press, ISBN 978-0-12-163251-9
Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
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