Product Topology

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## Definition

## Examples

## Properties

## Relation to other topological notions

## Axiom of choice

## See also

## Notes

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Product Topology

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**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

Given *X*, also known as the product space, such that

is the Cartesian product of the topological spaces *X _{i}*, indexed by , and the

The open sets in the product topology are unions (finite or infinite) of sets of the form , where each *U _{i}* is open in

The product topology on *X* is the topology generated by sets of the form *p _{i}*

In general, the product of the topologies of each *X _{i}* forms a basis for what is called the box topology on

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 **R**^{n}.

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.

Several additional examples are given in the article on the initial topology.

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*, *f _{i}* :

This shows that the product space is a product in the category of topological spaces. It follows from the above universal property that a map *f* : *Y* → *X* is continuous if and only if *f _{i}* =

In addition to being continuous, the canonical projections *p _{i}* :

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 *X*_{i} converge. In particular, if one considers the space *X* = **R**^{I} of all real valued functions on *I*, convergence in the product topology is the same as pointwise convergence of functions.

Any product of closed subsets of *X _{i}* is a closed set in

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.

- Separation
- Every product of T
_{0}spaces is T_{0} - Every product of T
_{1}spaces is T_{1} - Every product of Hausdorff spaces is Hausdorff
^{[1]} - Every product of regular spaces is regular
- Every product of Tychonoff spaces is Tychonoff
- A product of normal spaces
*need not*be normal

- Every product of T
- Compactness
- Every product of compact spaces is compact (Tychonoff's theorem)
- A product of locally compact spaces
*need not*be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact*is*locally compact (This condition is sufficient and necessary).

- Connectedness
- Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
- Every product of hereditarily disconnected spaces is hereditarily disconnected.

- Metric spaces
- Countable products of metric spaces are metrizable

One of many ways to express the axiom of choice is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.^{[2]} The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.

The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice,^{[3]} and shows why the product topology may be considered the more useful topology to put on a Cartesian product.

**^**"Product topology preserves the Hausdorff property".*PlanetMath*.**^**Pervin, William J. (1964),*Foundations of General Topology*, Academic Press, p. 33**^**Hocking, John G.; Young, Gail S. (1988) [1961],*Topology*, Dover, p. 28, ISBN 978-0-486-65676-2

- Willard, Stephen (1970).
*General Topology*. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796. Retrieved 2013.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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