Product topology: Difference between revisions
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The product topology on an arbitrary product <math>\textstyle\prod_{\lambda \in \Lambda} X_\lambda</math> is the topology with sub-basis <math>\textstyle\prod_{\lambda \in \Lambda} G_\lambda</math> where each ''G''<sub>λ</sub> is open in ''X''<sub>λ</sub> and where all but finitely many of the ''G''<sub>λ</sub> are equal to the whole of the corresponding ''X''<sub>λ</sub>. | The product topology on an arbitrary product <math>\textstyle\prod_{\lambda \in \Lambda} X_\lambda</math> is the topology with sub-basis <math>\textstyle\prod_{\lambda \in \Lambda} G_\lambda</math> where each ''G''<sub>λ</sub> is open in ''X''<sub>λ</sub> and where all but finitely many of the ''G''<sub>λ</sub> are equal to the whole of the corresponding ''X''<sub>λ</sub>. | ||
The product topology has a [[universal property]]: if there is a topological space ''Z'' with [[continuous map]]s <math>f_\lambda:Z \rightarrow X_\lambda</math>, then there is a continuous map <math>h : Z \ | The product topology has a [[universal property]]: if there is a topological space ''Z'' with [[continuous map]]s <math>f_\lambda:Z \rightarrow X_\lambda</math>, then there is a continuous map <math>\textstyle h : Z \to \prod_{\lambda \in \Lambda} X_\lambda</math> such that the compositions <math>h \cdot \mathrm{pr}_\lambda = f_\lambda</math>. This map ''h'' is defined by | ||
:<math> h(z) = ( \lambda \mapsto f_\lambda(z) ) . \, </math> | :<math> h(z) = ( \lambda \mapsto f_\lambda(z) ) . \, </math> | ||
The projection maps pr<sub>λ</sub> to the factor spaces are continuous and [[open map]]s. | The projection maps pr<sub>λ</sub> to the factor spaces are continuous and [[open map]]s. | ||
The product of two (and hence finitely many) [[compact space]]s is compact. | |||
The ''[[Tychonoff product theorem]]'': The product of any family of compact spaces is compact. | |||
==References== | ==References== | ||
* {{cite book | author=Wolfgang Franz | title=General Topology | publisher=Harrap | year=1967 | pages=52-55 }} | * {{cite book | author=Wolfgang Franz | title=General Topology | publisher=Harrap | year=1967 | pages=52-55 }} | ||
* {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=90-91}} | * {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=90-91}}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:00, 7 October 2024
In general topology, the product topology is an assignment of open sets to the Cartesian product of a family of topological spaces.
The product topology on a product of two topological spaces (X,T) and (Y,U) is the topology with sub-basis for open sets of the form G×H where G is open in X (that is, G is an element of T) and H is open in Y (that is, H is an element of U). So a set is open in the product topology if is a union of products of open sets.
By iteration, the product topology on a finite Cartesian product X1×...×Xn is the topology with sub-basis of the form G1×...×Gn.
The product topology on an arbitrary product is the topology with sub-basis where each Gλ is open in Xλ and where all but finitely many of the Gλ are equal to the whole of the corresponding Xλ.
The product topology has a universal property: if there is a topological space Z with continuous maps , then there is a continuous map such that the compositions . This map h is defined by
The projection maps prλ to the factor spaces are continuous and open maps.
The product of two (and hence finitely many) compact spaces is compact.
The Tychonoff product theorem: The product of any family of compact spaces is compact.
References
- Wolfgang Franz (1967). General Topology. Harrap, 52-55.
- J.L. Kelley (1955). General topology. van Nostrand, 90-91.