# Difference between revisions of "Restricted direct product"

(Start article: Restricted direct product) |
m (better) |
||

(One intermediate revision by the same user not shown) | |||

Line 1: | Line 1: | ||

− | {{TEX|done}}{{MSC|03E|54B10 | + | {{TEX|done}}{{MSC|03E|54B10,11R56}} |

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]]. | + | 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=34849