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,
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
let
be a
topological space.
Denote the
Cartesian product of the sets
by
and for every index
denote the
-th by
The , sometimes called the , on
is defined to be the
coarsest topology (that is, 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 .
The open sets in the product topology are arbitrary 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
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
A
subset of
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
The
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
converges if and only if all its projections to the spaces
converge.
Explicitly, a sequence
(respectively, a net
) converges to a given point
if and only if
in
for every index
where
denotes
(respectively, denotes
).
In particular, if
is used for all
then the Cartesian product is the space
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
and convergence in the product topology is the same as
pointwise convergence of functions.
Examples
If the
real line 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
copies of
is equal to the ordinary
Euclidean topology on
(Because
is finite, this is also equivalent to the
box topology on
)
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
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.
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 continuous map
such that for each
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
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 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:
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
is fixed then the set
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
.
Relation to other topological notions
Separation
* Every product of
T0 spaces is T
0.
* Every product of
T1 spaces is T
1.
* 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