Restricted direct product

In mathematics , the restricted direct product is a topological construction from the theory of locally compact groups .

It defines a topological space that is formed from a given family of topological spaces with the help of the Cartesian product . If the family is finite, the restricted direct product is the Cartesian product equipped with the product topology . In the case of infinite products, however, you generally get a different topology than the product topology.

In contrast to the product topology, the restricted direct product of locally compact spaces always delivers a locally compact space.

definition

Let be a family of topological spaces and be given an open compact subset for almost all of them . The restricted direct product X of (with respect to ) is the amount ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle i\in I}$ ${\displaystyle Y_{i}\subseteq X_{i}}$ ${\displaystyle X_{i}}$ ${\displaystyle Y_{i}}$

${\displaystyle X:=\{(x_{i})_{i\in I}\in \prod _{i\in I}X_{i}\mid {\text{für fast alle }}i\in I{\text{ gilt }}x_{i}\in Y_{i}\}}$

together with the following topology: An open rectangle in X is a subset of the shape

${\displaystyle R=\prod _{i\in I}U_{i},}$

where is open and equality applies to almost everyone . The intersection of a finite number of open rectangles is an open rectangle. A subset is called open if it can be written as a union of open rectangles, so the open rectangles form a basis for the topology on X. ${\displaystyle U_{i}\subseteq X_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle U_{i}=Y_{i}}$ ${\displaystyle U\subseteq X}$

Write

${\displaystyle X={\widehat {\prod _{i\in I}}}^{Y_{i}}X_{i}}$

for the restricted direct product. If it is clear which amount to choose, write briefly: ${\displaystyle Y_{i}}$

${\displaystyle X={\widehat {\prod _{i\in I}}}X_{i}}$

Further define for a finite subset of ${\displaystyle S}$ ${\displaystyle I}$

${\displaystyle X_{S}=\prod _{i\in S}X_{i}\times \prod _{i\notin S}Y_{i}.}$

Then there is an open subset of and the subspace topology of on is equal to the product topology of on . ${\displaystyle X_{S}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X_{S}}$ ${\displaystyle X_{S}}$

Note: It applies . ${\displaystyle X=\bigcup \limits _{S\subset I{\text{ endlich}}}X_{S}}$

properties

• The topology induced by the product topology is coarser. That is, any subset of the constrained direct product that is open to the topology induced by the product topology is open.
• If all are locally compact and those are compact, then X is also locally compact. The direct product, on the other hand, is locally compact if, in addition, almost all of them are compact. ${\displaystyle X_{i}}$ ${\displaystyle Y_{i}}$ ${\displaystyle X_{i}}$
• The restricted product topology depends on the totality of , but not on the individual , i.e. H. be open to everyone and it applies to almost everyone . Then the two constrained direct products and their corresponding topologies are canonically isomorphic. ${\displaystyle X_{i}}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{i}'\subset X_{i}}$ ${\displaystyle i}$ ${\displaystyle X_{i}'=X_{i}}$ ${\displaystyle i}$
• The restricted direct product can be decomposed as follows. Let be a disjoint decomposition of the index set . Then: ${\displaystyle I=I_{1}{\overset {.}{\cup }}I_{2}}$ ${\displaystyle I}$

${\displaystyle {\widehat {\prod _{i\in I_{1}}}}^{Y_{i}}X_{i}\times {\widehat {\prod _{i\in I_{2}}}}^{Y_{i}}X_{i}={\widehat {\prod _{i\in I}}}^{Y_{i}}X_{i}}$

where the two sets and the two topologies match (on the left we have the product topology for the two factors). The proof of this statement is not difficult: Note that the same set is defined on both sides and the same open sets generate the respective topology, so the two topological spaces are the same.

• If they are even locally compact groups, then we can fix corresponding dimensions on them . We can normalize these dimensions so that is. Then define the product dimension by setting it to restricted open rectangles. Since these generate the topology, it is sufficient to define on them. So define ${\displaystyle X_{i}}$ ${\displaystyle \mu _{i}}$ ${\displaystyle \mu _{i}(Y_{i})=1}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$

${\displaystyle \mu (\prod \limits _{i\in E}U_{i}\times \prod \limits _{i\in I\setminus E}Y_{i}):=\prod \limits _{i\in E}\mu _{i}(U_{i})\cdot \prod \limits _{i\in I\setminus E}\mu _{i}(Y_{i})=\prod \limits _{i\in E}\mu _{i}(U_{i}).}$

This is a finite product that is and is finite. ${\displaystyle \mu _{i}(Y_{i})=1}$ ${\displaystyle E}$

Examples

• Is for almost everyone , you get the product topology . ${\displaystyle Y_{i}=X_{i}}$ ${\displaystyle i}$
• The ring of Adele is the restricted direct product of with regard to the (for we simply do not take an open subset of , according to the definition it is sufficient if such a subset is given for almost all p). ${\displaystyle \mathbb {A} }$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle p=\infty }$ ${\displaystyle \mathbb {Q} _{p}=\mathbb {R} }$
• The group of idels is the restricted direct product of those relating to . Note that the topology on does not match the subspace topology induced by . ${\displaystyle \mathbb {A} ^{\times }}$ ${\displaystyle \mathbb {Q} _{p}^{\times }}$ ${\displaystyle \mathbb {Z} _{p}^{\times }}$ ${\displaystyle \mathbb {A} ^{\times }}$ ${\displaystyle \mathbb {A} }$

literature

• Anton Deitmar: Automorphic forms . Springer, Berlin Heidelberg 2010, ISBN 978-3-642-12389-4 , page 122f.
• John Cassels , Albrecht Froehlich: Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute) . Academic Press, London 1967, XVIII, 366 pages.