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| 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 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 [[product (category theory)|categorical product]] of its factors, whereas the box topology is [[Comparison of topologies|too fine]]; this is the sense in which the product topology is "natural".
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| ==Definition==
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| Given ''X'' such that
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| :<math>X := \prod_{i \in I} X_i,</math>
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| is the Cartesian product of the topological spaces ''X<sub>i</sub>'', [[index set|indexed]] by <math>i \in I</math>, and the '''[[projection (set theory)|canonical projections]]''' ''p<sub>i</sub>'' : ''X'' → ''X<sub>i</sub>'', the '''product topology''' on ''X'' is defined to be the [[coarsest topology]] (i.e. the topology with the fewest open sets) for which all the projections ''p<sub>i</sub>'' are [[continuous (topology)|continuous]]. The product topology is sometimes called the '''Tychonoff topology'''.
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| The open sets in the product topology are unions (finite or infinite) of sets of the form <math>\prod_{i\in I} U_i</math>, where each ''U<sub>i</sub>'' is open in ''X<sub>i</sub>'' and ''U''<sub>''i''</sub> ≠ ''X''<sub>''i''</sub> for only finitely many ''i''. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''X<sub>i</sub>'' gives a basis for the product <math>\prod_{i\in I} X_i</math>.
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| The product topology on ''X'' is the topology generated by sets of the form ''p<sub>i</sub>''<sup>−1</sup>(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''X<sub>i</sub>''. In other words, the sets {''p<sub>i</sub>''<sup>−1</sup>(''U'')} form a [[subbase]] for the topology on ''X''. A [[subset]] of ''X'' is open if and only if it is a (possibly infinite) [[union (set theory)|union]] of [[intersection (set theory)|intersections]] of finitely many sets of the form ''p<sub>i</sub>''<sup>−1</sup>(''U''). The ''p<sub>i</sub>''<sup>−1</sup>(''U'') are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.
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| In general, the product of the topologies of each ''X<sub>i</sub>'' forms a basis for what is called the [[box topology]] on ''X''. In general, the box topology is [[finer topology|finer]] than the product topology, but for finite products they coincide.
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| == Examples ==
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| If one starts with the [[standard topology]] on the [[real line]] '''R''' and defines a topology on the product of ''n'' copies of '''R''' in this fashion, one obtains the ordinary [[Euclidean topology]] on '''R'''<sup>''n''</sup>.
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| The [[Cantor set]] is [[homeomorphic]] to the product of [[countable|countably many]] copies of the [[discrete space]] {0,1} 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.
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| Several additional examples are given in the article on the [[initial topology]].
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| == Properties ==
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| 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<sub>i</sub>'' : ''Y'' → ''X<sub>i</sub>'' is a continuous map, then there exists ''precisely one'' continuous map ''f'' : ''Y'' → ''X'' such that for each ''i'' in ''I'' the following diagram [[commutative diagram|commutes]]:
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| [[Image:CategoricalProduct-02.png|center|Characteristic property of product spaces]]
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| 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 ''f'' : ''Y'' → ''X'' is continuous [[if and only if]] ''f<sub>i</sub>'' = ''p<sub>i</sub>'' o ''f'' is continuous for all ''i'' in ''I''. In many cases it is easier to check that the component functions ''f<sub>i</sub>'' are continuous. Checking whether a map ''g'' : ''X''→ ''Z'' is continuous is usually more difficult; one tries to use the fact that the ''p<sub>i</sub>'' are continuous in some way.
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| In addition to being continuous, the canonical projections ''p<sub>i</sub>'' : ''X'' → ''X<sub>i</sub>'' are [[open map]]s. This means that any open subset of the product space remains open when projected down to the ''X<sub>i</sub>''. The converse is not true: if ''W'' is a [[subspace (topology)|subspace]] of the product space whose projections down to all the ''X<sub>i</sub>'' are open, then ''W'' need not be open in ''X''. (Consider for instance ''W'' = '''R'''<sup>2</sup> \ (0,1)<sup>2</sup>.) The canonical projections are not generally [[closed map]]s (consider for example the closed set <math>\{(x,y) \in \mathbb{R}^2 \mid xy = 1\},</math> whose projections onto both axes are '''R''' \ {0}).
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| The product topology is also called the ''topology of pointwise convergence'' because of the following fact: a [[sequence]] (or [[Net (mathematics)|net]]) in ''X'' converges if and only if all its projections to the spaces ''X''<sub>''i''</sub> converge. In particular, if one considers the space ''X'' = '''R'''<sup>''I''</sup> of all [[real number|real]] valued [[function (mathematics)|function]]s on ''I'', convergence in the product topology is the same as pointwise convergence of functions.
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| Any product of closed subsets of ''X<sub>i</sub>'' is a closed set in ''X''.
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| An important theorem about the product topology is [[Tychonoff's theorem]]: any product of [[compact space]]s is compact. This is easy to show for finite products, while the general statement is equivalent to the [[axiom of choice]].
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| == Relation to other topological notions ==
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| * Separation
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| ** Every product of [[T0 space|T<sub>0</sub> space]]s is T<sub>0</sub>
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| ** Every product of [[T1 space|T<sub>1</sub> space]]s is T<sub>1</sub>
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| ** Every product of [[Hausdorff space]]s is Hausdorff<ref>{{planetmath reference|id=4317|title=Product topology preserves the Hausdorff property}}</ref>
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| ** Every product of [[regular space]]s is regular
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| ** Every product of [[Tychonoff space]]s is Tychonoff
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| ** A product of [[normal space]]s ''need not'' be normal
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| * Compactness
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| ** Every product of compact spaces is compact ([[Tychonoff's theorem]])
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| ** A product of [[locally compact space]]s ''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).
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| * Connectedness
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| ** Every product of [[connectedness|connected]] (resp. path-connected) spaces is connected (resp. path-connected)
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| ** Every product of hereditarily disconnected spaces is hereditarily disconnected.
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| ==Axiom of choice==
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| The [[axiom of choice]] is equivalent to the statement that the product of a collection of non-empty sets is non-empty. The proof is easy enough: 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.
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| 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.
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| == See also ==
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| *[[Disjoint union (topology)]]
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| *[[Initial topology|Projective limit topology]]
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| *[[Quotient space]]
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| *[[Subspace (topology)]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{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 |accessdate=13 February 2013}}
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| ==External links==
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| * {{planetmath reference|id=3100|title=product topology}}
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| [[Category:Topology]]
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| [[Category:General topology]]
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| [[Category:Binary operations]]
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Hello from United States. I'm glad to be here. My first name is Mittie.
I live in a city called Dekalb in south United States.
I was also born in Dekalb 26 years ago. Married in January 1999. I'm working at the the office.
my blog post: Fifa 15 Coin Hack