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×...×Xn is the topology with sub-basis of the form G1×...×Gn.
The product topology on an arbitrary product is the topology with sub-basis where each Gλ is open in Xλ and where all but finitely many of the Gλ are equal to the whole of the corresponding Xλ.
The projection maps prλ 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.
- Wolfgang Franz (1967). General Topology. Harrap, 52-55.
- J.L. Kelley (1955). General topology. van Nostrand, 90-91.