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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 natural-seeming, 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, I will be some non-empty
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \left(x_j\right)_ &&\;\mapsto\;& x_i \\ \end The , sometimes called the , on \prod_ X_i is defined to be the coarsest topology (that is, the topology with the fewest open sets) for which all the projections p_i : \prod X_ \to X_i are continuous. The Cartesian product X := \prod_ X_i endowed with the product topology is called the . The open sets in the product topology are arbitrary unions (finite or infinite) of sets of the form \prod_ U_i, where each U_i is open in X_i and U_i \neq X_i for only finitely many i. 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 X_i gives a basis for the product topology of \prod_ X_i. That is, for a finite product, the set of all \prod_ U_i, where U_i is an element of the (chosen) basis of X_i, is a basis for the product topology of \prod_ X_i. The product topology on \prod_ X_i is the topology generated by sets of the form p_i^\left(U_i\right), where i \in I and U_i is an open subset of X_i. In other words, the sets \left\ form a
subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
for the topology on X. A subset of X is open if and only if it is a (possibly infinite)
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of intersections of finitely many sets of the form p_i^\left(U_i\right). The p_i^\left(U_i\right) are sometimes called
open cylinder In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod_ ...
s, and their intersections are cylinder sets. The product topology is also called the because a sequence (or more generally, a
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
) in \prod_ X_i converges if and only if all its projections to the spaces X_i converge. Explicitly, a sequence s_ = \left(s_n\right)_^ (respectively, a net s_ = \left(s_a\right)_) converges to a given point x \in \prod_ X_i if and only if p_i\left(s_\right) \to p_i(x) in X_i for every index i \in I, where p_i\left(s_\right) := p_i \circ s_ denotes \left(p_i\left(s_n\right)\right)_^ (respectively, denotes \left(p_i\left(s_a\right)\right)_). In particular, if X_i = \R is used for all i then the Cartesian product is the space \prod_ \R = \R^I of all
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s on I, and convergence in the product topology is the same as pointwise convergence of functions.


Examples

If the real line \R is endowed with its
standard topology In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
then the product topology on the product of n copies of \R is equal to the ordinary Euclidean topology on \R^n. (Because n is finite, this is also equivalent to the box topology on \R^n.) The Cantor set is homeomorphic to the product of countably many copies of the discrete space \ and the space of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s 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 In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
.


Properties

The set of Cartesian products between the open sets of the topologies of each X_i 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. 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 : Y \to X_i is a continuous map, then there exists continuous map f : Y \to X such that for each i \in I the following diagram commutes. This shows that the product space is a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
in the category of topological spaces. It follows from the above universal property that a map f : Y \to X is continuous if and only if f_i = p_i \circ f is continuous for all i \in I. In many cases it is easier to check that the component functions f_i are continuous. Checking whether a map X \to Y is continuous is usually more difficult; one tries to use the fact that the p_i are continuous in some way. In addition to being continuous, the canonical projections p_i : X \to X_i are open maps. This means that any open subset of the product space remains open when projected down to the X_i. The converse is not true: if W is a subspace of the product space whose projections down to all the X_i are open, then W need not be open in X (consider for instance W = \R^2 \setminus (0, 1)^2.) The canonical projections are not generally closed maps (consider for example the closed set \left\, whose projections onto both axes are \R \setminus \). Suppose \prod_ S_i is a product of arbitrary subsets, where S_i \subseteq X_i for every i \in I. If all S_i are then \prod_ S_i is a closed subset of the product space X if and only if every S_i is a closed subset of X_i. More generally, the closure of the product \prod_ S_i of arbitrary subsets in the product space X is equal to the product of the closures: \operatorname_X \left(\prod_ S_i\right) = \prod_ \left(\operatorname_ S_i\right). Any product of
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s is again a Hausdorff space.
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
, which is equivalent to the axiom of choice, states that any product of compact spaces is a compact space. A specialization of
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
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 z = \left(z_i\right)_ \in X is fixed then the set \left\ is a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of the product space X.


Relation to other topological notions

Separation * Every product of T0 spaces is T0. * Every product of T1 spaces is T1. * Every product of
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s is Hausdorff. * Every product of regular spaces is regular. * Every product of
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
s is Tychonoff. * A product of
normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
s be normal. Compactness * Every product of compact spaces is compact (
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
). * A product of
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
s be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact locally compact (This condition is sufficient and necessary). Connectedness * Every product of
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
(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 space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
s.


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. 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 In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
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, and shows why the product topology may be considered the more useful topology to put on a Cartesian product.


See also

* * * - Sometimes called the projective limit topology * * * * *


Notes


References

* * {{cite book, last=Willard , first=Stephen , title=General Topology , year=1970 , publisher=Addison-Wesley Pub. Co. , location=Reading, Mass. , isbn=0486434796 , url=http://store.doverpublications.com/0486434796.html , access-date=13 February 2013 General topology Operations on structures