View source for Product topology

{{Short description|Topology on Cartesian products of topological spaces}}
{{Redirect|Product space||The Product Space}}
In [[topology]] and related areas of [[mathematics]], a '''product space''' is the [[Cartesian product]] of a family of [[topological space]]s 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 [[Comparison of topologies|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 [[Product (category theory)|categorical product]] of its factors, whereas the box topology is too [[Comparison of topologies|fine]]; in that sense the product topology is the natural topology on the Cartesian product.

==Definition==
Throughout, <math>I</math> will be some non-empty [[index set]] and for every index <math>i \in I,</math> let <math>X_i</math> be a [[topological space]].
Denote the [[Cartesian product]] of the sets <math>X_i</math> by

<math display=block>X := \prod X_{\bull} := \prod_{i \in I} X_i</math>

and for every index <math>i \in I</math>, denote the <math>i</math>-th '''{{em|[[Projection (set theory)|canonical projection]]}}''' by

<math display=block>\begin{align}
p_i :\ \prod_{j \in I} X_j &\to     X_i, \\[3mu]
       (x_j)_{j \in I}     &\mapsto x_i. \\
\end{align}</math>

The '''{{em|{{visible anchor|product topology}}}}''', sometimes called the '''{{em|{{visible anchor|Tychonoff topology}}}}''', on <math display="inline">\prod_{i \in I} X_i</math> is defined to be the [[coarsest topology]] (that is, the topology with the fewest open sets) for which all the projections <math display="inline">p_i : \prod X_{\bull} \to X_i</math> are [[Continuous (topology)|continuous]]. It is the [[initial topology]] on <math display="inline">\prod_{i \in I} X_i</math> with respect to the family of projections <math>\left\{p_i\mathbin{\big\vert} i \in I\right\}</math>. The Cartesian product <math display="inline">X := \prod_{i \in I} X_i</math> endowed with the product topology is called the '''{{em|{{visible anchor|product space}}}}'''.
The open sets in the product topology are the unions of (finitely many or infinitely many) sets of the form <math display="inline">\prod_{i \in I} U_i</math>, where each <math>U_i</math> is open in <math>X_i</math> and <math>U_i \neq X_i</math> for only finitely many <math>i</math>. 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 <math>X_i</math> gives a basis for the product topology of <math display="inline">\prod_{i\in I} X_i</math>. That is, for a finite product, the set of all <math display="inline">\prod_{i \in I} U_i</math>, where each <math>U_i</math> is an element of the (chosen) basis of <math>X_i</math>, is a basis for the product topology of <math display="inline">\prod_{i\in I} X_i</math>.

The product topology on <math display="inline">\prod_{i \in I} X_i</math> is the topology [[Base (topology)|generated]] by the sets of the form <math>p_i^{-1}\left(U\right)</math>, where <math>i \in I</math> and <math>U</math> is an open subset of <math>X_i</math>. A [[subset]] of <math>X</math> is open if and only if it is the [[Union (set theory)|union]] of (possibly infinitely many) [[Intersection (set theory)|intersections]] of finitely many sets of the form <math>p_i^{-1}\left(U\right)</math>. The <math>p_i^{-1}\left(U\right)</math>'s are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.

The product topology is also called the {{em|[[topology of pointwise convergence]]}} because a [[sequence]] (or more generally, a [[Net (mathematics)|net]]) in <math display="inline">\prod_{i \in I} X_i</math> converges if and only if all its projections to the spaces <math>X_i</math> converge.
Explicitly, a sequence <math display="inline">s_{\bull} = \left(s_n\right)_{n=1}^{\infty}</math> (respectively, a net <math display="inline">s_{\bull} = \left(s_a\right)_{a \in A}</math>) converges to a given point <math display="inline">x \in \prod_{i \in I} X_i</math> if and only if <math>p_i\left(s_{\bull}\right) \to p_i(x)</math> in <math>X_i</math>  for every index <math>i \in I</math>, where <math>p_i\left(s_{\bull}\right) := p_i \circ s_{\bull}</math> denotes <math>\left(p_i\left(s_n\right)\right)_{n=1}^{\infty}</math> (respectively, denotes <math>\left(p_i\left(s_a\right)\right)_{a \in A}</math>).
In particular, if <math>X_i = \R</math> for all <math>i</math>, then the Cartesian product is the space <math display="inline">\prod_{i \in I} \R = \R^I</math> of all [[Real number|real]]-valued [[Function (mathematics)|function]]s on <math>I</math>, and convergence in the product topology is the same as [[pointwise convergence]] of functions.

==Examples==

If the [[real line]] <math>\R</math> is endowed with its [[standard topology]] then the product topology on the product of <math>n</math> copies of <math>\R</math> is equal to the ordinary [[Euclidean topology]] on <math>\R^n.</math> (Because <math>n</math> is finite, this is also equivalent to the [[box topology]] on <math>\R^n.</math>)

The [[Cantor set]] is [[homeomorphic]] to the product of [[Countable|countably many]] copies of the [[discrete space]] <math>\{ 0, 1 \}</math> and the space of [[irrational number]]s is homeomorphic to the product of countably many copies of the [[natural number]]s, where again each copy carries the discrete topology.

Several additional examples are given in the article on the [[initial topology]].

==Properties==

The set of Cartesian products between the open sets of the topologies of each <math>X_i</math> forms a basis for what is called the [[box topology]] on <math>X.</math> In general, the box topology is [[Finer topology|finer]] than the product topology, but for finite products they coincide.

The product space <math>X,</math> together with the canonical projections, can be characterized by the following [[universal property]]: if <math>Y</math> is a topological space, and for every <math>i \in I,</math> <math>f_i : Y \to X_i</math> is a continuous map, then there exists {{em|precisely one}} continuous map <math>f : Y \to X</math> such that for each <math>i \in I</math> the following diagram [[Commutative diagram|commutes]]:

{{bi|left=1.3em|[[File:CategoricalProduct-02.svg|frameless|upright=0.6|Characteristic property of product spaces|class=skin-invert]] }}

This shows that the product space is a [[Product (category theory)|product]] in the [[category of topological spaces]]. It follows from the above universal property that a map <math>f : Y \to X</math> is continuous [[if and only if]] <math>f_i = p_i \circ f</math> is continuous for all <math>i \in I.</math> In many cases it is easier to check that the component functions <math>f_i</math> are continuous. Checking whether a map <math> X \to Y</math> is continuous is usually more difficult; one tries to use the fact that the <math>p_i</math> are continuous in some way.

In addition to being continuous, the canonical projections <math>p_i : X \to X_i</math> are [[open map]]s. This means that any open subset of the product space remains open when projected down to the <math>X_i.</math> The converse is not true: if <math>W</math> is a [[Subspace (topology)|subspace]] of the product space whose projections down to all the <math>X_i</math> are open, then <math>W</math> need not be open in <math>X</math> (consider for instance <math display="inline">W = \R^2 \setminus (0, 1)^2.</math>) The canonical projections are not generally [[closed map]]s (consider for example the closed set <math display="inline">\left\{(x,y) \in \R^2 : xy = 1\right\},</math> whose projections onto both axes are <math>\R \setminus \{0\}</math>).

Suppose <math display="inline">\prod_{i \in I} S_i</math> is a product of arbitrary subsets, where <math>S_i \subseteq X_i</math> for every <math>i \in I.</math> If all <math>S_i</math> are {{em|non-empty}}  then <math display="inline">\prod_{i \in I} S_i</math> is a closed subset of the product space <math>X</math> if and only if every <math>S_i</math> is a closed subset of <math>X_i.</math> More generally, the closure of the product <math display="inline">\prod_{i \in I} S_i</math> of arbitrary subsets in the product space <math>X</math> is equal to the product of the closures:{{sfn|Bourbaki|1989|pp=43-50}}

<math display=block>
{\operatorname{Cl}_X}\Bigl(\prod_{i \in I} S_i\Bigr) = \prod_{i \in I} \bigl({\operatorname{Cl}_{X_i}} S_i\bigr).
</math>

Any product of [[Hausdorff space]]s is again a Hausdorff space; this comes from a simple preservation of disjoint open neighborhoods under the inverse of canonical projections from <math>X_i</math>'s to <math display="inline">\prod_{i \in I} X_i</math>.

[[Tychonoff's theorem]], which is equivalent to the [[axiom of choice]], states that any product of [[compact space]]s 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 space|Hausdorff]] spaces is a compact space.

If <math display="inline">z = \left(z_i\right)_{i \in I} \in X</math> is fixed then the set

<math display=block>
\left\{ x = \left(x_i\right)_{i \in I} \in X \mathbin{\big\vert} x_i = z_i \text{ for all but finitely many } i \right\}
</math>

is a [[Dense set|dense subset]] of the product space <math>X</math>.{{sfn|Bourbaki|1989|pp=43-50}}

==Relation to other topological notions==

'''Separation'''
* Every product of [[T0 space|T<sub>0</sub> space]]s is T<sub>0</sub>.
* Every product of [[T1 space|T<sub>1</sub> space]]s is T<sub>1</sub>.
* Every product of [[Hausdorff space]]s is Hausdorff.
* Every product of [[regular space]]s is regular.
* Every product of [[Tychonoff space]]s is Tychonoff.
* A product of [[normal space]]s {{em|need not}} be normal.

'''Compactness'''
* Every product of compact spaces is compact ([[Tychonoff's theorem]]).
* A product of [[locally compact space]]s {{em|need not}} be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact {{em|is}} locally compact (This condition is sufficient and necessary).

'''Connectedness'''
* Every product of [[Connectedness|connected]] (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]]s are [[metrizable space]]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.<ref>{{citation|first=William J.|last=Pervin|title=Foundations of General Topology|year=1964|publisher=Academic Press|page=33}}</ref> 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,<ref>{{citation|first1=John G.|last1=Hocking|first2=Gail S.|last2=Young|title=Topology|year=1988|orig-year=1961|publisher=Dover|isbn=978-0-486-65676-2|page=[https://archive.org/details/topology00hock_0/page/28 28]|url=https://archive.org/details/topology00hock_0/page/28}}</ref> and shows why the product topology may be considered the more useful topology to put on a Cartesian product.

==See also==
* {{annotated link|Disjoint union (topology)}}
* {{annotated link|Final topology}}
* {{annotated link|Initial topology}} - Sometimes called the projective limit topology
* {{annotated link|Inverse limit}}
* {{annotated link|Pointwise convergence}}
* {{annotated link|Quotient space (topology)}}
* {{annotated link|Subspace (topology)}}
* {{annotated link|Weak topology}}

==Notes==
{{reflist}}

==References==
* {{Bourbaki General Topology Part I Chapters 1-4}} <!-- {{sfn|Bourbaki|1989|p=}} -->
* {{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}}
* {{Cite book |last=Kelley |first=John L. |title=General Topology |publisher=[[Dover Publications]] |year=2017 |isbn=978-0-486-81544-2 |edition=Dover}}

[[Category:General topology]]
[[Category:Operations on structures]]