In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related fields of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sequential space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak
axiom of countability, and all
first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
s (especially
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s) are sequential.
In any topological space
if a convergent sequence is contained in a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
then the
limit of that sequence must be contained in
as well. This property is known as sequential closure. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of
nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.
Any topology can be
refined (that is, made finer) to a sequential topology, called the sequential coreflection of
The related concepts of
Fréchet–Urysohn spaces, -sequential spaces, and
-sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.
Sequential spaces and
-sequential spaces were introduced by
S. P. Franklin.
History
Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is originally due to S. P. Franklin in 1965. Franklin wanted to determine "the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences", and began by investigating the
first-countable spaces, for which it was already known that sequences sufficed. Franklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.
Preliminary definitions
Let
be a set and let
be a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
in
; that is, a family of elements of
,
indexed by the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
. In this article,
means that each element in the sequence
is an element of
and, if
is a map, then
For any index
the tail of
starting at
is the sequence
A sequence
is eventually in
if some tail of
satisfies
Let
be a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on
and
a sequence therein. The sequence
converges to a point
written
(when context allows,
), if, for every neighborhood
of
eventually
is in
is then called a limit point of
A function
between topological spaces is
sequentially continuous if
implies
Sequential closure/interior
Let
be a topological space and let
be a subset. The
topological closure (resp.
topological interior) of
in
is denoted by
(resp.
).
The sequential closure of
in
is the set
which defines a map, the sequential closure operator, on the power set of
If necessary for clarity, this set may also be written
or
It is always the case that
but the reverse may fail.
The sequential interior of
in
is the set
(the topological space again indicated with a subscript if necessary).
Sequential closure and interior satisfy many of the nice properties of ''topological'' closure and interior: for all subsets
- and ;
- and ;
- ;
- ; and
That is, sequential closure is a
preclosure operator. Unlike topological closure, sequential closure is not
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
: the last containment may be strict. Thus sequential closure is not a (
Kuratowski)
closure operator.
Sequentially closed and open sets
A set
is sequentially closed if
; equivalently, for all
and
such that
we must have
[You cannot simultaneously apply this "test" to infinitely many subsets (for example, you can not use something akin to the ]axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
). Not all sequential spaces are Fréchet-Urysohn, but only in those spaces can the closure of a set can be determined without it ever being necessary to consider any set other than
A set
is defined to be sequentially open if its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
is sequentially closed. Equivalent conditions include:
- or
- For all and such that eventually is in (that is, there exists some integer such that the tail ).
A set
is a sequential neighborhood of a point
if it contains
in its sequential interior; sequential neighborhoods need ''not'' be sequentially open (see below).
It is possible for a subset of
to be sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.
Sequential spaces and coreflection
As discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology. One can obtain an idempotent sequential closure via
transfinite iteration: for a
successor ordinal define (as usual)
and, for a
limit ordinal define
This process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index
(the
first uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. Wh ...
). Conversely, the sequential order of
is the minimal ordinal at which, for any choice of
the above sequence will stabilize.
The transfinite sequential closure of
is the terminal set in the above sequence:
The operator
is idempotent and thus a
closure operator. In particular, it defines a topology, the sequential coreflection. In the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).
Sequential spaces
A topological space
is sequential if it satisfies any of the following equivalent conditions:
- is its own sequential coreflection.
- Every sequentially open subset of is open.
- Every sequentially closed subset of is closed.
- For any subset that is closed in there exists some
[A Fréchet–Urysohn space is defined by the analogous condition for all such : ]For any subset that is not closed in ''for any'' there exists a sequence in that converges to
and a sequence in that converges to [ Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12 ]
- (Universal Property) For every topological space a map is continuous if and only if it is sequentially continuous (if then ).
- is the quotient of a first-countable space.
- is the quotient of a metric space.
By taking
and
to be the identity map on
in the universal property, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences. If two topologies agree on convergent sequences, then they necessarily have the same sequential coreflection. Moreover, a function from
is sequentially continuous if and only if it is continuous on the sequential coreflection (that is, when pre-composed with
).
- and -sequential spaces
A -sequential space is a topological space with sequential order 1, which is equivalent to any of the following conditions:
- The sequential closure (or interior) of every subset of is sequentially closed (resp. open).
- or are idempotent.
- or
- Any sequential neighborhood of can be shrunk to a sequentially-open set that contains ; formally, sequentially-open neighborhoods are a neighborhood basis for the sequential neighborhoods.
- For any and any sequential neighborhood of there exists a sequential neighborhood of such that, for every the set is a sequential neighborhood of
Being a -sequential space is incomparable with being a sequential space; there are sequential spaces that are not -sequential and vice-versa. However, a topological space
is called a
-sequential (or neighborhood-sequential) if it is both sequential and -sequential. An equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.
Every
first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
(and thus every
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) ...
) is
-sequential. There exist
topological vector spaces that are sequential but
-sequential (and thus not -sequential).
Fréchet–Urysohn spaces
A topological space
is called
Fréchet–Urysohn if it satisfies any of the following equivalent conditions:
- is hereditarily sequential; that is, every topological subspace is sequential.
- For every subset
- For any subset that is not closed in and every there exists a sequence in that converges to
Fréchet–Urysohn spaces are also sometimes said to be "Fréchet," but should be confused with neither
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to t ...
s in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
nor the
T1 condition.
Examples and sufficient conditions
Every
CW-complex is sequential, as it can be considered as a quotient of a metric space.
The
prime spectrum of a commutative
Noetherian ring with the
Zariski topology is sequential.
Take the real line
and
identify the set
of integers to a point. As a quotient of a metric space, the result is sequential, but it is not first countable.
Every
first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential. Thus every metrizable or
pseudometrizable space — in particular, every
second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \m ...
,
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, or
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
— is sequential.
Let
be a set of maps from
Fréchet–Urysohn spaces to
Then the
final topology that
induces on
is sequential.
A Hausdorff
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is sequential if and only if there exists no strictly finer topology with the same convergent sequences.
[Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507]
Spaces that are sequential but not Fréchet-Urysohn
Schwartz space and the space
of
smooth functions, as discussed in the article on
distributions, are both widely-used sequential spaces, but are not
Fréchet-Urysohn. Indeed the
strong dual spaces of both these of spaces are not
Fréchet-Urysohn either.
[T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.]
More generally, every infinite-dimensional
Montel DF-space is sequential but not
Fréchet–Urysohn.
Arens' space is sequential, but not Fréchet–Urysohn.
Non-examples (spaces that are not sequential)
The simplest space that is not sequential is the
cocountable topology on an uncountable set. Every convergent sequence in such a space is eventually constant; hence every set is sequentially open. But the cocountable topology is not
discrete. (One could call the topology "sequentially discrete".)
Let
denote the
space of -smooth test functions with its canonical topology and let
denote the space of distributions, the
strong dual space of
; neither are sequential (nor even an
Ascoli space).
On the other hand, both
and
are
Montel spaces
and, in the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of any Montel space, a ''sequence'' of continuous linear functionals converges in the
strong dual topology if and only if it converges in the
weak* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
(that is, converges pointwise).
Consequences
Every sequential space has
countable tightness and is
compactly generated.
If
is a continuous
open surjection
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
between two Hausdorff sequential spaces then the set
of points with unique preimage is closed. (By continuity, so is its preimage in
the set of all points on which
is injective.)
If
is a surjective map (not necessarily continuous) onto a Hausdorff sequential space
and
bases for the topology on
then
is an
open map if and only if, for every
basic neighborhood
of
and sequence
in
there is a subsequence of
that is eventually in
Categorical properties
The
full subcategory Seq of all sequential spaces is closed under the following operations in the
category Top of topological spaces:
The category Seq is closed under the following operations in Top:
Since they are closed under topological sums and quotients, the sequential spaces form a
coreflective subcategory of the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again conti ...
. In fact, they are the coreflective hull of
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 (that is, the smallest class of topological spaces closed under sums and quotients and containing the metrizable spaces).
The subcategory Seq is a
Cartesian closed category with respect to its own product (not that of Top). The
exponential objects are equipped with the (convergent sequence)-open topology.
P.I. Booth and A. Tillotson have shown that Seq is the smallest Cartesian closed subcategory of Top containing the underlying topological spaces of all
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s,
CW-complexes, and
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s and that is closed under colimits, quotients, and other "certain reasonable identities" that
Norman Steenrod
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology.
Life
He was born in Dayton, Ohio, and educated at Miami University and University o ...
described as "convenient".
.
Every sequential space is
compactly generated, and finite products in Seq coincide with those for compactly generated spaces, since products in the category of compactly generated spaces preserve quotients of metric spaces.
See also
*
*
*
*
*
Notes
Citations
References
* Arkhangel'skii, A.V. and Pontryagin, L.S., ''General Topology I'', Springer-Verlag, New York (1990) .
*
*
*
*
* Engelking, R., ''General Topology'', Heldermann, Berlin (1989). Revised and completed edition.
*
*
*
* Goreham, Anthony,
Sequential Convergence in Topological Spaces, (2016)
*
*
*
*
*
* {{Wilansky Modern Methods in Topological Vector Spaces, edition=1
General topology
Properties of topological spaces