Product topology

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In general topology, the product topology is an assignment of open sets to the Cartesian product of a familiy of topological space.

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 prodct 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 arbitary product $\prod_{\lambda \in \Lambda} X_\lambda$ is the topology with sub-basis $\prod_{\lambda \in \Lambda} G_\lambda$ where each G_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 $h : Z \rightarrow \prod_{\lambda \in \Lambda} X_\lambda$ such that the compositions $h \cdot \mathrm{pr}_\lambda = f_\lambda$ . This map h is defined by