Product topology

TheInfoList

OR:

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a product space is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of a family of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s equipped with a
natural topology In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
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 In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or ring ...
of its factors, whereas the box topology is too
fine Fine may refer to: Characters * Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny'' * Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano Legal terms * Fine (penalty), money to be paid as punishment for an offe ...
; 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
. Denote the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of the sets $X_i$ by $X := \prod X_ := \prod_ X_i$ and for every index $i \in I,$ denote the $i$-th by The , sometimes called the , on $\prod_ X_i$ is defined to be the
coarsest topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as t ...
(that is, the topology with the fewest open sets) for which all the projections $p_i : \prod X_ \to X_i$ are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** 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 $p_i^\left\left(U_i\right\right),$ 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 $p_i^\left\left(U_i\right\right).$ The $p_i^\left\left(U_i\right\right)$ 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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
(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 $p_i\left\left(s_\right\right) \to p_i\left(x\right)$ in $X_i$ for every index $i \in I,$ where $p_i\left\left(s_\right\right) := p_i \circ s_$ denotes $\left\left(p_i\left\left(s_n\right\right)\right\right)_^$ (respectively, denotes $\left\left(p_i\left\left(s_a\right\right)\right\right)_$). 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 In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
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 In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, 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'' tha ...
.

# 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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
: 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
$f_i = p_i \circ f$ 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 map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
s. 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 map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
s (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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, states that any product of
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s 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 space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' ca ...
s 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 space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s 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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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.