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Difference between revisions of "Restricted direct product"

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A generalisation of the [[direct product]] of a family of sets $X_\lambda$ ranging over an index set $\Lambda$ where there are given subsets $Y_\lambda \subseteq X_\lambda$ for all but finitely many $\lambda \in \Lambda$.  The elements of the restricted direct product are those $(x_\lambda) : x_\lambda \in X_\lambda, \ \lambda \in \Lambda$ for which $x_\lambda \in Y_\lambda$ for all but finitely many $\lambda \in \Lambda$ (the exceptional $\lambda$ including those for which no such $Y_\lambda$ is specified).
 
A generalisation of the [[direct product]] of a family of sets $X_\lambda$ ranging over an index set $\Lambda$ where there are given subsets $Y_\lambda \subseteq X_\lambda$ for all but finitely many $\lambda \in \Lambda$.  The elements of the restricted direct product are those $(x_\lambda) : x_\lambda \in X_\lambda, \ \lambda \in \Lambda$ for which $x_\lambda \in Y_\lambda$ for all but finitely many $\lambda \in \Lambda$ (the exceptional $\lambda$ including those for which no such $Y_\lambda$ is specified).
  
If the $X_\lambda$ are [[pointed set]]s with basepoints $y_\lambda \in X_\lambda$, then we take $Y_\lambda = \{ y_\lambda \}$ for all $\lambda \in \Lambda$ and the restricted direct product becomes the [[direct sum]].
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If the $X_\lambda$ are [[pointed set]]s with basepoints $y_\lambda \in X_\lambda$, then we may take $Y_\lambda = \{ y_\lambda \}$ for all $\lambda \in \Lambda$ and then the restricted direct product becomes the [[direct sum]].
  
 
An important special case is when the $X_\lambda$ are topological Abelian groups or rings (cf. [[Topological group]], [[Topological ring]]).  Assume that the $X_\lambda$ are [[Locally compact space|locally compact]] and that the $Y_\lambda$ are compact neighbourhoods of zero.  We define a topology on the restricted direct product by taking as a basis of neighbourhoods of zero  those sets $\prod_\lambda U_\lambda$ with $U_\lambda$ open in $X_\lambda$ for all $\lambda \in \Lambda$ and $U_\lambda = Y_\lambda$ for all but finitely many $\lambda$.  The restricted product topology is again locally compact.  This is the construction of the topology on the [[Idèle]] and [[Adele group]]s of a [[global field]].
 
An important special case is when the $X_\lambda$ are topological Abelian groups or rings (cf. [[Topological group]], [[Topological ring]]).  Assume that the $X_\lambda$ are [[Locally compact space|locally compact]] and that the $Y_\lambda$ are compact neighbourhoods of zero.  We define a topology on the restricted direct product by taking as a basis of neighbourhoods of zero  those sets $\prod_\lambda U_\lambda$ with $U_\lambda$ open in $X_\lambda$ for all $\lambda \in \Lambda$ and $U_\lambda = Y_\lambda$ for all but finitely many $\lambda$.  The restricted product topology is again locally compact.  This is the construction of the topology on the [[Idèle]] and [[Adele group]]s of a [[global field]].

Latest revision as of 08:22, 1 January 2017

2010 Mathematics Subject Classification: Primary: 03E Secondary: 54B1011R56 [MSN][ZBL]

A generalisation of the direct product of a family of sets $X_\lambda$ ranging over an index set $\Lambda$ where there are given subsets $Y_\lambda \subseteq X_\lambda$ for all but finitely many $\lambda \in \Lambda$. The elements of the restricted direct product are those $(x_\lambda) : x_\lambda \in X_\lambda, \ \lambda \in \Lambda$ for which $x_\lambda \in Y_\lambda$ for all but finitely many $\lambda \in \Lambda$ (the exceptional $\lambda$ including those for which no such $Y_\lambda$ is specified).

If the $X_\lambda$ are pointed sets with basepoints $y_\lambda \in X_\lambda$, then we may take $Y_\lambda = \{ y_\lambda \}$ for all $\lambda \in \Lambda$ and then the restricted direct product becomes the direct sum.

An important special case is when the $X_\lambda$ are topological Abelian groups or rings (cf. Topological group, Topological ring). Assume that the $X_\lambda$ are locally compact and that the $Y_\lambda$ are compact neighbourhoods of zero. We define a topology on the restricted direct product by taking as a basis of neighbourhoods of zero those sets $\prod_\lambda U_\lambda$ with $U_\lambda$ open in $X_\lambda$ for all $\lambda \in \Lambda$ and $U_\lambda = Y_\lambda$ for all but finitely many $\lambda$. The restricted product topology is again locally compact. This is the construction of the topology on the Idèle and Adele groups of a global field.

References

  • Algebraic Number Theory, 2nd ed. J.W.S. Cassels, A. Fröhlich (edd) London Mathematical Society (2010) ISBN 0950273422. First ed. Zbl 0645.12001
How to Cite This Entry:
Restricted direct product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Restricted_direct_product&oldid=34852