Product topology

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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.

DefinitionEdit

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.

ExamplesEdit

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

PropertiesEdit

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:

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 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 notionsEdit

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 choiceEdit

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 is equivalent to the axiom of choice,[4] and shows why the product topology may be considered the more useful topology to put on a Cartesian product.

See alsoEdit

NotesEdit

  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

ReferencesEdit