Product topology

Summary

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.

Definition

Throughout, will be some non-empty index set and for every index will be a topological space. Let

be the Cartesian product of the sets and denote the canonical projections by The product topology, sometimes called the Tychonoff topology, on is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections are continuous. The Cartesian product endowed with the product topology is called the product space. The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (or net) in converges if and only if all its projections to the spaces converge. In particular, if one considers the space of all real valued functions on convergence in the product topology is the same as pointwise convergence of functions.

The open sets in the product topology are unions (finite or infinite) of sets of the form where each is open in and for only finitely many In particular, for a finite product (in particular, for the product of two topological spaces), the set of all Cartesian products between one basis element from each gives a basis for the product topology of That is, for a finite product, the set of all where is an element of the (chosen) basis of is a basis for the product topology of

The product topology on is the topology generated by sets of the form where and is an open subset of In other words, the sets

form a subbase for the topology on A subset of is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form The are sometimes called open cylinders, and their intersections are cylinder sets.

The product of the topologies of each forms a basis for what is called the box topology on In general, the box topology is finer than the product topology, but for finite products they coincide.

Examples

If the real line is endowed with its standard topology then the product topology on the product of copies of is equal to the ordinary Euclidean topology on

The Cantor set is homeomorphic to the product of countably many copies of the discrete space 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.

Properties

The product space together with the canonical projections, can be characterized by the following universal property: if is a topological space, and for every is a continuous map, then there exists precisely one continuous map such that for each the following diagram commutes.

Characteristic property of product spaces

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 is continuous if and only if is continuous for all In many cases it is easier to check that the component functions are continuous. Checking whether a map is continuous is usually more difficult; one tries to use the fact that the are continuous in some way.

In addition to being continuous, the canonical projections are open maps. This means that any open subset of the product space remains open when projected down to the The converse is not true: if is a subspace of the product space whose projections down to all the are open, then need not be open in (consider for instance ) The canonical projections are not generally closed maps (consider for example the closed set whose projections onto both axes are ).

Suppose is a product of arbitrary subsets, where for every If all are non-empty then is a closed subset of the product space if and only if every is a closed subset of More generally, the closure of the product of arbitrary subsets in the product space is equal to the product of the closures:[1]

Any product of Hausdorff spaces is again a Hausdorff space.

Tychonoff's theorem, which is equivalent to the axiom of choice, states any product of compact spaces is a compact space. A specialization of Tychonoff's theorem that requires only the ultrafilter lemma (and not the full strength of the axiom of choice) states that any product of compact Hausdorff spaces is a compact space.

If is fixed then the set

is a dense subset of the product space .[1]

Relation to other topological notions

Separation
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

Axiom of choice

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.[3] 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 requires the axiom of choice and is equivalent to it in its most general formulation,[4] and shows why the product topology may be considered the more useful topology to put on a Cartesian product.

See also

Notes

  1. ^ a b Bourbaki 1989, pp. 43–50.
  2. ^ "Product topology preserves the Hausdorff property". PlanetMath.
  3. ^ Pervin, William J. (1964), Foundations of General Topology, Academic Press, p. 33
  4. ^ Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 28, ISBN 978-0-486-65676-2

References

  • Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
  • Willard, Stephen (1970). General Topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796. Retrieved 13 February 2013.