# Product topology

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In topology, the cartesian product of topological space 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 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 (possibly infinte) union of finite intersections of sets of the form pi-1(O), where i in I and O is an open subset of Xi.

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 exist precisely one continuous map f : Y -> X such that pi o f = fi for all i in I.

If one defines a topology on the product of n copies of the real numbers R in this fashion, one obtains the ordinary euclidean topology on Rn.

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.

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.