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Product topology

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In topology, the cartesian product of topological spaces is turned into a topological space in the following way. Let I be a (possibly infinite) index set and suppose Xi is a topological space for every i in I. Set X = Π Xi, the cartesian product of the sets Xi. For every i in I, we have a canonical projection pi : X -> Xi. The product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) which turns all the maps pi into continuous maps.

Explicitly, the topology on X can be described as follows. A subset of X is open if and only if it is a union of (possibly infinitely many) intersections of finitely many sets of the form pi-1(O), where i in I and O is an open subset of Xi. This implies that, in general, not all products of open sets need to be open in X.

Examples

If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn.

The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology.

Properties

The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (or net) in X converges if and only if all its projections to the spaces Xi converge. In particular, if one considers the space X = RI of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions.

An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. This is easy for finite products, but the statement is (surprisingly) also true for infinite products, when the proof requires the axiom of choice in some form.

The product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, and for every i in I, fi : Y -> Xi is a continuous map, then there exists precisely one continuous map f : Y -> X such that pi o f = fi for all i in I. This shows that the product space is a product in the sense of category theory.

To check whether a given map f : Y -> X is continuous, one can use the following handy criterion: f is continuous if and only if pi o f is continuous for all i in I. Checking whether a map g : X -> Z is continuous is usually more difficult; one tries to use the fact that the pi are continuous in some way.

Relation to other topological notions

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