Product topology: Difference between revisions

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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×...×X_n is the topology with sub-basis of the form G₁×...×G_n.

The product topology on an arbitrary product $\textstyle \prod _{\lambda \in \Lambda }X_{\lambda }$ is the topology with sub-basis $\textstyle \prod _{\lambda \in \Lambda }G_{\lambda }$ where each G_λ is open in X_λ and where all but finitely many of the G_λ are equal to the whole of the corresponding X_λ.

The product topology has a universal property: if there is a topological space Z with continuous maps $f_{\lambda }:Z\rightarrow X_{\lambda }$ , then there is a continuous map $\textstyle h:Z\to \prod _{\lambda \in \Lambda }X_{\lambda }$ such that the compositions $h\cdot \mathrm {pr} _{\lambda }=f_{\lambda }$ . This map h is defined by

h(z)=(\lambda \mapsto f_{\lambda }(z)).\,

The projection maps pr_λ 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.

@@ Line 1: / Line 1: @@
-In [[general topology]], the '''product topology''' is an assignment of open sets to the [[Cartesian product]] of a familiy of [[topological space]].
+{{subpages}}
+In [[general topology]], the '''product topology''' is an assignment of open sets to the [[Cartesian product]] of a family of [[topological space]]s.
 The product topology on a product of two topological spaces (''X'',''T'') and (''Y'',''U'') is the topology with [[Sub-basis (topology)|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 prodct topology on a finite Cartesian product ''X''<sub>1</sub>×...×''X''<sub>''n''</sub> is the topology with sub-basis of the form ''G''<sub>1</sub>×...×''G''<sub>''n''</sub>.
+By iteration, the product topology on a finite Cartesian product ''X''<sub>1</sub>×...×''X''<sub>''n''</sub> is the topology with sub-basis of the form ''G''<sub>1</sub>×...×''G''<sub>''n''</sub>.
-The product topology on an arbitary product <math>\prod_{\lambda \in \Lambda} X_\lambda</math> is the topology with sub-basis <math>\prod_{\lambda \in \Lambda} G_\lambda</math> where each ''G''<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 \rightarrow \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
+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>
 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==
 * {{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]]

Product topology: Difference between revisions

Latest revision as of 11:00, 7 October 2024

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