Sequential space explained

In topology and related fields of mathematics, a sequential space is a topological space 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 spaces (notably metric spaces) are sequential.

In any topological space

(X,\tau),

if a convergent sequence is contained in a closed set

C,

then the limit of that sequence must be contained in

C

as well. Sets with this property are known as sequentially closed. 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

X.

The related concepts of Fréchet–Urysohn spaces, -sequential spaces, and

N

-sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.

Sequential spaces and

N

-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 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

See also: Filters in topology and Net (mathematics).

Let

X

be a set and let

x\bull=\left(xi\right)

infty
i=1
be a sequence in

X

; that is, a family of elements of

X

, indexed by the natural numbers. In this article,

x\bull\subseteqS

means that each element in the sequence

x\bull

is an element of

S,

and, if

f:X\toY

is a map, then

f\left(x\bull\right)=\left(f\left(xi\right)\right)

infty
i=1

.

For any index

i,

the tail of

x\bull

starting at

i

is the sequence x_ = (x_i, x_, x_, \ldots)\text A sequence

x\bull

is eventually in

S

if some tail of

x\bull

satisfies

x\geq\subseteqS.

Let

\tau

be a topology on

X

and

x\bull

a sequence therein. The sequence

x\bull

converges to a point

x\inX,

written

x\bull\overset{\tau}{\to}x

(when context allows,

x\bull\tox

), if, for every neighborhood

U\in\tau

of

x,

eventually

x\bull

is in

U.

x

is then called a limit point of

x\bull.

A function

f:X\toY

between topological spaces is sequentially continuous if

x\bull\tox

implies

f(x\bull)\tof(x).

Sequential closure/interior

Let

(X,\tau)

be a topological space and let

S\subseteqX

be a subset. The topological closure (resp. topological interior) of

S

in

(X,\tau)

is denoted by

\operatorname{cl}XS

(resp.

\operatorname{int}XS

).

The sequential closure of

S

in

(X,\tau)

is the set\operatorname(S) = \left\which defines a map, the sequential closure operator, on the power set of

X.

If necessary for clarity, this set may also be written

\operatorname{scl}X(S)

or

\operatorname{scl}(X,\tau)(S).

It is always the case that

\operatorname{scl}XS\subseteq\operatorname{cl}XS,

but the reverse may fail.

The sequential interior of

S

in

(X,\tau)

is the set\operatorname(S) = \(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

R,S\subseteqX,

That is, sequential closure is a preclosure operator. Unlike topological closure, sequential closure is not idempotent: the last containment may be strict. Thus sequential closure is not a (Kuratowski) closure operator.

Sequentially closed and open sets

A set

S

is sequentially closed if

S=\operatorname{scl}(S)

; equivalently, for all

s\bull\subseteqS

and

x\inX

such that

s\bull\overset{\tau}{\to}x,

we must have

x\inS.

[1]

A set

S

is defined to be sequentially open if its complement is sequentially closed. Equivalent conditions include:

A set

S

is a sequential neighborhood of a point

x\inX

if it contains

x

in its sequential interior; sequential neighborhoods need not be sequentially open (see below).

It is possible for a subset of

X

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

\alpha+1,

define (as usual)(\operatorname)^(S)=\operatorname((\operatorname)^\alpha(S))and, for a limit ordinal

\alpha,

define(\operatorname)^\alpha(S)=\bigcup_\textThis process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index

\omega1

(the first uncountable ordinal). Conversely, the sequential order of

X

is the minimal ordinal at which, for any choice of

S,

the above sequence will stabilize.[2]

The transfinite sequential closure of

S

is the terminal set in the above sequence:
\omega1
(\operatorname{scl})

(S).

The operator
\omega1
(\operatorname{scl})
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).[3]

Sequential spaces

A topological space

(X,\tau)

is sequential if it satisfies any of the following equivalent conditions:

By taking

Y=X

and

f

to be the identity map on

X

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

Y

is sequentially continuous if and only if it is continuous on the sequential coreflection (that is, when pre-composed with

f

).

- and -sequential spaces

A -sequential space is a topological space with sequential order 1, which is equivalent to any of the following conditions:[8]

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

(X,\tau)

is called a

N

-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 (and thus every metrizable space) is

N

-sequential. There exist topological vector spaces that are sequential but

N

-sequential (and thus not -sequential).

Fréchet–Urysohn spaces

See main article: Fréchet–Urysohn space.

A topological space

(X,\tau)

is called Fréchet–Urysohn if it satisfies any of the following equivalent conditions:

Fréchet–Urysohn spaces are also sometimes said to be "Fréchet," but should be confused with neither Fréchet spaces in functional analysis 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.[9]

Take the real line

\R

and identify the set

\Z

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 is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential. Thus every metrizable or pseudometrizable space  - in particular, every second-countable space, metric space, or discrete space  - is sequential.

Let

l{F}

be a set of maps from Fréchet–Urysohn spaces to

X.

Then the final topology that

l{F}

induces on

X

is sequential.

A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences.[10]

Spaces that are sequential but not Fréchet-Urysohn

l{S}\left(\Rn\right)

and the space

Cinfty(U)

of smooth functions, as discussed in the article on distributions, are both widely-used sequential spaces.[11] [12]

More generally, every infinite-dimensional Montel DF-space is sequential but not Fréchet–Urysohn.

Arens' space is sequential, but not Fréchet–Urysohn.[13] [14]

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".)[15]

Let

k(U)
C
c
denote the space of

k

-smooth test functions with its canonical topology and let

l{D}'(U)

denote the space of distributions, the strong dual space of
infty
C
c

(U)

; neither are sequential (nor even an Ascoli space). On the other hand, both
infty
C
c

(U)

and

l{D}'(U)

are Montel spaces[16] and, in the dual space 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 (that is, converges pointwise).

Consequences

Every sequential space has countable tightness and is compactly generated.

If

f:X\toY

is a continuous open surjection between two Hausdorff sequential spaces then the set

\{y:{|f-1(y)|=1}\}\subseteqY

of points with unique preimage is closed. (By continuity, so is its preimage in

X,

the set of all points on which

f

is injective.)

If

f:X\toY

is a surjective map (not necessarily continuous) onto a Hausdorff sequential space

Y

and

l{B}

bases for the topology on

X,

then

f:X\toY

is an open map if and only if, for every

x\inX,

basic neighborhood

B\inl{B}

of

x,

and sequence

y\bull=\left(yi\right)

infty
i=1

\tof(x)

in

Y,

there is a subsequence of

y\bull

that is eventually in 

f(B).

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 fact, they are the coreflective hull of metrizable spaces (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 spaces, CW-complexes, and differentiable manifolds and that is closed under colimits, quotients, and other "certain reasonable identities" that Norman Steenrod 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.

References

Notes and References

  1. You cannot simultaneously apply this "test" to infinitely many subsets (for example, you can not use something akin to the axiom of choice). Not all sequential spaces are Fréchet-Urysohn, but only in those spaces can the closure of a set

    S

    can be determined without it ever being necessary to consider any set other than

    S.

    • Arhangel'skiĭ . A. V. . Franklin . S. P. . 1968 . Ordinal invariants for topological spaces. . Michigan Math. J. . 15 . 3 . 313–320 . 10.1307/mmj/1029000034 . free.
  2. Baron . S. . October 1968 . The Coreflective Subcategory of Sequential Spaces . . en . 11 . 4 . 603–604 . 10.4153/CMB-1968-074-4 . 124685527 . 0008-4395. free .
  3. Web site: Topology of sequentially open sets is sequential? . Mathematics Stack Exchange.
  4. A Fréchet–Urysohn space is defined by the analogous condition for all such

    x

    :
    For any subset

    S\subseteqX

    that is not closed in

    X,

    for any

    x\in\operatorname{cl}X(S)\setminusS,

    there exists a sequence in

    S

    that converges to

    x.

  5. Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12
  6. Baron . S. . Leader . Solomon . 1966 . Solution to Problem #5299 . The American Mathematical Monthly . 73 . 6 . 677–678 . 10.2307/2314834 . 2314834 . 0002-9890.
  7. Snipes . Ray . 1972 . T-sequential topological spaces . Fundamenta Mathematicae . en . 77 . 2 . 95–98 . 10.4064/fm-77-2-95-98 . 0016-2736.
  8. Web site: 2004 . On sequential properties of Noetherian topological spaces . 30 Jul 2023.
  9. Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507
  10. 1702.07867. Gabrielyan . Saak. Topological properties of strict

    (LF)

    -spaces and strong duals of Montel strict

    (LF)

    -spaces. Monatshefte für Mathematik. 189. 1. 91–99. 2019. 10.1007/s00605-018-1223-6.
  11. T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.
  12. Engelking 1989, Example 1.6.19
  13. Web site: Ma . Dan . 19 August 2010 . A note about the Arens' space . 1 August 2013.
  14. Web site: math . Sleziak . Martin . Dec 6, 2016 . Example of different topologies with same convergent sequences . 2022-06-27 . Mathematics Stack Exchange . StackOverflow . en.
  15. Web site: . Topological vector space . September 6, 2020 . Encyclopedia of Mathematics . "It is a Montel space, hence paracompact, and so normal.".