Product topology

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×...×Xn is the topology with sub-basis of the form G1×...×Gn.

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 Gλ 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 $$f_\lambda:Z \rightarrow X_\lambda$$, 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


 * $$ h(z) = ( \lambda \mapsto f_\lambda(z) ) . \, $$

The projection maps prλ to the factor spaces are continuous and open maps.