Product topology

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.