# Product topology

(Redirected from Product space)

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

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## Definition

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 : XXi. The product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous.

Explicitly, the product topology on X can be described as the topology generated by sets of the form pi−1(U), where i in I and U is an open subset of Xi. In other words, the sets {pi−1(U)} form a subbase for the topology on X. 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(U).

We can describe a basis for the product topology using bases of the constituting spaces Xi. Suppose that for each i in I we choose a set Yi which is either the whole space Xi or a basis set of that space, in such a way that Xi = Yi for all but finitely many i in I. Let B be the cartesian product of the sets Yi. The collection of all sets B that can be constructed in this fashion is a basis of the product space. In particular, this means that a product of finitely many spaces has a basis given by the products of base elements of the Xi.

If the index set is finite (in particular, for a product of two topological spaces) then the product topology admits a simpler description. In this case product of the topologies of each Xi forms a basis for the topology on X. In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, but for finite products they coincide.

## 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 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 : YXi is a continuous map, then there exists precisely one continuous map f : YX such that the following diagram commutes:

Missing image
CategoricalProduct-02.png
Characteristic property of product spaces

This shows that the product space is a product in the category of topological spaces. If follows from the above universal property that a map f : YX is continuous iff fi = pi o f is continuous for all i in I. In many cases it is often easier to check that the component functions fi are continuous. Checking whether a map g : XZ is continuous is usually more difficult; one tries to use the fact that the pi are continuous in some way.

In addition to being continuous, the canonical projections pi : XXi are open maps. This means that any open subset of the product space remains open when projected down to the Xi. The converse is not true: if W is a subspace of the product space whose projections down to all the Xi are open, then W need not be open in X. (Consider for instance W = R2 \ (0,1)2.) The canonical projections are not generally closed maps.

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 to show for finite products, while the general statement is equivalent to the axiom of choice.

## Relation to other topological notions

A map that "locally looks like" a canonical projection F × UU is called a fiber bundle.

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