Product topology

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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×...×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 product topology has a universal property: if there is a topological space Z with continuous maps , then there is a continuous map such that the compositions . This map h is defined by

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.

References

  • Wolfgang Franz (1967). General Topology. Harrap, 52-55. 
  • J.L. Kelley (1955). General topology. van Nostrand, 90-91.