Editing Sequence (section)

==Use in other fields of mathematics==

===Topology===
Sequences play an important role in topology, especially in the study of [[metric spaces]]. For instance:
* A [[metric space]] is [[compact space|compact]] exactly when it is [[sequential compactness|sequentially compact]].
* A function from a metric space to another metric space is [[continuous function|continuous]] exactly when it takes convergent sequences to convergent sequences.
* A metric space is a [[connected space]] if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.
* A [[topological space]] is [[separable space|separable]] exactly when there is a dense sequence of points.

Sequences can be generalized to [[Net (mathematics)|nets]] or [[Filter (set theory)|filters]]. These generalizations allow one to extend some of the above theorems to spaces without metrics.

====Product topology====
The [[product topology|topological product]] of a sequence of topological spaces is the [[cartesian product]] of those spaces, equipped with a [[natural topology]] called the [[product topology]].

More formally, given a sequence of spaces <math>(X_i)_{i\in\mathbb N}</math>, the product space

:<math>X := \prod_{i\in\mathbb N} X_i, </math>

is defined as the set of all sequences <math>(x_i)_{i\in\mathbb N}</math> such that for each ''i'', <math>x_i</math> is an element of <math>X_i</math>. The '''[[projection (set theory)|canonical projections]]''' are the maps ''p<sub>i</sub>'' : ''X'' &rarr; ''X<sub>i</sub>'' defined by the equation <math>p_i((x_j)_{j\in\mathbb N}) = x_i</math>. Then 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'''.

===Analysis===
When discussing sequences in [[mathematical analysis|analysis]], one will generally consider sequences of the form
:<math>(x_1, x_2, x_3, \dots)\text{ or }(x_0, x_1, x_2, \dots)</math>
which is to say, infinite sequences of elements indexed by [[natural number]]s.

A sequence may start with an index different from 1 or 0. For example, the sequence defined by ''x<sub>n</sub>'' = 1/[[logarithm|log]](''n'') would be defined only for ''n'' ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices [[large enough]], that is, greater than some given ''N''.

The most elementary type of sequences are numerical ones, that is, sequences of [[real number|real]] or [[complex number|complex]] numbers. This type can be generalized to sequences of elements of some [[vector space]]. In analysis, the vector spaces considered are often [[function space]]s. Even more generally, one can study sequences with elements in some [[topological space]].

====Sequence spaces====
{{main|Sequence space}}
A [[sequence space]] is a [[vector space]] whose elements are infinite sequences of [[real number|real]] or [[complex number|complex]] numbers. Equivalently, it is a [[function space]] whose elements are functions from the [[natural numbers]] to the [[Field (mathematics)|field]] ''K'', where ''K'' is either the field of real numbers or the field of complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in ''K'', and can be turned into a [[vector space]] under the operations of [[pointwise addition]] of functions and pointwise scalar multiplication. All sequence spaces are [[linear subspace]]s of this space. Sequence spaces are typically equipped with a [[norm (mathematics)|norm]], or at least the structure of a [[topological vector space]].

The most important sequences spaces in analysis are the ℓ<sup>''p''</sup> spaces, consisting of the ''p''-power summable sequences, with the ''p''-norm. These are special cases of [[Lp space|L<sup>''p''</sup> spaces]] for the [[counting measure]] on the set of natural numbers. Other important classes of sequences like convergent sequences or [[Sequence_space#c,_c0_and_c00|null sequence]]s form sequence spaces, respectively denoted ''c'' and ''c''<sub>0</sub>, with the sup norm. Any sequence space can also be equipped with the [[topology]] of [[pointwise convergence]], under which it becomes a special kind of [[Fréchet space]] called an [[FK-space]].

===Linear algebra===
Sequences over a [[field (mathematics)|field]] may also be viewed as [[Vector (geometric)|vectors]] in a [[vector space]]. Specifically, the set of ''F''-valued sequences (where ''F'' is a field) is a [[function space]] (in fact, a [[product space]]) of ''F''-valued functions over the set of natural numbers.

===Abstract algebra===
Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings.

====Free monoid====
{{Main|Free monoid}}
If ''A'' is a set, the [[free monoid]] over ''A'' (denoted ''A''<sup>*</sup>, also called [[Kleene star]] of ''A'') is a [[monoid]] containing all the finite sequences (or strings) of zero or more elements of ''A'', with the binary operation of concatenation. The [[free semigroup]] ''A''<sup>+</sup> is the subsemigroup of ''A''<sup>*</sup> containing all elements except the empty sequence.

====Exact sequences====
{{Main|Exact sequence}}
In the context of [[group theory]], a sequence
:<math>G_0 \;\overset{f_1}{\longrightarrow}\; G_1 \;\overset{f_2}{\longrightarrow}\; G_2 \;\overset{f_3}{\longrightarrow}\; \cdots \;\overset{f_n}{\longrightarrow}\; G_n</math>
of [[group (mathematics)|groups]] and [[group homomorphism]]s is called '''exact''', if the [[Image (mathematics)|image]] (or [[Range of a function|range]]) of each homomorphism is equal to the [[Kernel (algebra)|kernel]] of the next:
:<math>\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1})</math>

The sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other [[algebraic structure]]s. For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of [[module (mathematics)|modules]] and [[module homomorphism]]s.

====Spectral sequences====
{{Main|Spectral sequence}}
In [[homological algebra]] and [[algebraic topology]], a '''spectral sequence''' is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of [[exact sequence]]s, and since their introduction by {{harvs|txt|authorlink=Jean Leray|first=Jean|last=Leray|year=1946}}, they have become an important research tool, particularly in [[homotopy theory]].

===Set theory===
An [[Order topology#Ordinal-indexed sequences|ordinal-indexed sequence]] is a generalization of a sequence. If α is a [[limit ordinal]] and ''X'' is a set, an α-indexed sequence of elements of ''X'' is a function from α to ''X''. In this terminology an ω-indexed sequence is an ordinary sequence.

===Computing===
In [[computer science]], finite sequences are called [[list (computer science)|lists]]. Potentially infinite sequences are called [[stream (computer science)|streams]]. Finite sequences of characters or digits are called [[String (computer science)|string]]s.

===Streams===
Infinite sequences of [[numerical digit|digits]] (or [[character (computing)|characters]]) drawn from a [[finite set|finite]] [[alphabet (computer science)|alphabet]] are of particular interest in [[theoretical computer science]]. They are often referred to simply as ''sequences'' or ''[[Stream (computing)|streams]]'', as opposed to finite ''[[String (computer science)#Formal theory|strings]]''. Infinite binary sequences, for instance, are infinite sequences of [[bit]]s (characters drawn from the alphabet {0, 1}). The set ''C'' = {0, 1}<sup>∞</sup> of all infinite binary sequences is sometimes called the [[Cantor space]].

An infinite binary sequence can represent a [[formal language]] (a set of strings) by setting the ''n''&thinsp;th bit of the sequence to 1 if and only if the ''n''&thinsp;th string (in [[shortlex order]]) is in the language. This representation is useful in the [[Cantor's diagonal argument|diagonalization method]] for proofs.<ref name=Oflazer2011>{{cite web|last1=Oflazer|first1=Kemal|title=FORMAL LANGUAGES, AUTOMATA AND COMPUTATION: DECIDABILITY|url=http://www.andrew.cmu.edu/user/ko/pdfs/lecture-15.pdf|website=cmu.edu|publisher=Carnegie-Mellon University|access-date=24 April 2015|archive-date=29 May 2015|archive-url=https://web.archive.org/web/20150529101719/http://www.andrew.cmu.edu/user/ko/pdfs/lecture-15.pdf|url-status=live}}</ref>