Product Topology

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

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

 R×...×R_()_(n times),

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

In the definition of product topology of X=product_(i in I)X_i, where I is any set, the open sets are the unions of subsets product_(i in I)U_i, where U_i is an open subset of X_i with the additional condition that U_i=X_i for all but finitely many indices i (this is automatically fulfilled if I 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 ith factor p_i:X->X_i continuous for all indices i. Admitting all products of open sets would give rise to a larger topology (strictly larger if I 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.

See also

Cantor's Discontinuum, Cartesian Product, Cube, Hilbert Cube, Productive Property, Product Metric, Product Space, Tychonoff Plank, Tychonoff Theorem

This entry contributed by Margherita Barile

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Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, pp. 65-91, 1968.Joshi, K. D. "Product Topology." §8.2 in Introduction to General Topology. New Delhi, India: Wiley, pp. 196-203, 1983.McCarty, G. "Tychonoff for Two." In Topology, an Introduction with Application to Topological Groups. New York: McGraw-Hill, pp. 154-157, 1967.Willard, S. "Product Spaces, Weak Topologies." §8 in General Topology. Reading, MA: Addison-Wesley, pp. 52-59, 1970.

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Product Topology

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Barile, Margherita. "Product Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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