Topology Sequences play an important role in topology, especially in the study of
metric spaces. For instance: • A
metric space is
compact exactly when it is
sequentially compact. • A function from a metric space to another metric space is
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 exactly when there is a dense sequence of points. Sequences can be generalized to
nets or
filters. These generalizations allow one to extend some of the above theorems to spaces without metrics.
Product topology The
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 (X_i)_{i\in\N}, the product space :X := \prod_{i\in\N} X_i, is defined as the set of all sequences (x_i)_{i\in\N} such that for each , x_i is an element of X_i. The
canonical projections are the maps defined by the equation p_i((x_j)_{j\in\N}) = x_i. Then the
product topology on is defined to be the
coarsest topology (i.e. the topology with the fewest open sets) for which all the projections are
continuous. The product topology is sometimes called the
Tychonoff topology.
Analysis When discussing sequences in
analysis, one will generally consider sequences of the form :(x_1, x_2, x_3, \dots)\text{ or }(x_0, x_1, x_2, \dots) which is to say, infinite sequences of elements indexed by
natural numbers. A sequence may start with an index different from or . For example, the sequence defined by , where is the
natural logarithm, would be defined only for . 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 . The most elementary type of sequences are numerical ones, that is, sequences of
real or
complex numbers. This type can be generalized to sequences of elements of some
vector space. In analysis, the vector spaces considered are often
function spaces. Even more generally, one can study sequences with elements in some
topological space.
Sequence spaces A
sequence space is a
vector space whose elements are infinite sequences of
real or
complex numbers. Equivalently, it is a
function space whose elements are functions from the
natural numbers to the
field , where 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 , 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 subspaces of this space. Sequence spaces are typically equipped with a
norm, or at least the structure of a
topological vector space. The most important sequences spaces in analysis are the spaces, consisting of the -power summable sequences, with the -norm. These are special cases of
spaces for the
counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or
null sequences form sequence spaces, respectively denoted and , 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 may also be viewed as
vectors in a
vector space. Specifically, the set of -valued sequences (where is a field) is a
function space (in fact, a
product space) of -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 If is a set, the
free monoid over (denoted , also called
Kleene star of ) is a
monoid containing all the finite sequences (or strings) of zero or more elements of , with the binary operation of concatenation. The
free semigroup is the subsemigroup of containing all elements except the empty sequence.
Exact sequences In the context of
group theory, a sequence :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 of
groups and
group homomorphisms is called
exact, if the
image (or
range) of each homomorphism is equal to the
kernel of the next: :\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1}) The sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for certain other
algebraic structures. For example, one could have an exact sequence of
vector spaces and
linear maps, or of
modules and
module homomorphisms.
Spectral sequences 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 sequences, and since their introduction by , they have become an important research tool, particularly in
homotopy theory.
Set theory An
ordinal-indexed sequence is a generalization of a sequence. If is a
limit ordinal and is a set, an -indexed sequence of elements of is a function from to . In this terminology an -indexed sequence is an ordinary sequence.
Computing In
computer science, finite sequences are called
lists. Potentially infinite sequences are called
streams. Finite sequences of characters or digits are called
strings.
Streams Infinite sequences of
digits (or
characters) drawn from a
finite alphabet are of particular interest in
theoretical computer science. They are often referred to simply as
sequences or
streams, as opposed to finite
strings. Infinite binary sequences, for instance, are infinite sequences of
bits (characters drawn from the alphabet ). The set {{math|1=
C = {0, 1}}} 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 th bit of the sequence to if and only if the th string (in
shortlex order) is in the language. This representation is useful in the
diagonalization method for proofs. ==See also==