Product Topology

The topology on the Cartesian product of two topological spaces whose open sets are the unions of subsets , where and are open subsets of and , respectively.

This definition extends in a natural way to the Cartesian product of any finite number of topological spaces. The product topology of

where is the real line with the Euclidean topology, coincides with the Euclidean topology of the Euclidean space .

In the definition of product topology of , where is any set, the open sets are the unions of subsets , where is an open subset of with the additional condition that for all but finitely many indices (this is automatically fulfilled if is a finite set). The reason for this choice of open sets is that these are the least needed to make the projection onto the th factor continuous for all indices . Admitting all products of open sets would give rise to a larger topology (strictly larger if is infinite), called the box topology.

The product topology is also called Tychonoff topology, but this should not cause any confusion with the notion of Tychonoff space, which has a completely different meaning.