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Sequential Convergence in Topological Spaces

, (2016) * * * * * * {{Wilansky Modern Methods in Topological Vector Spaces, edition=1 General topology Properties of topological spaces

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 $(X,\; \backslash tau),$ 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 ...

$C,$ then the limit of that sequence must be contained in $C$ 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 $X.$
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 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 $X$ be a set and let $x\_\; =\; \backslash left(x\_i\backslash right)\_^$ be asequence
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 $X$; that is, a family of elements of $X$, 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, $x\_\; \backslash subseteq\; S$ means that each element in the sequence $x\_$ is an element of $S,$ and, if $f\; :\; X\; \backslash to\; Y$ is a map, then $f\backslash left(x\_\backslash right)\; =\; \backslash left(f\backslash left(x\_i\backslash right)\backslash right)\_^.$ For any index $i,$ the tail of $x\_$ starting at $i$ is the sequence $$x\_\; =\; (x\_i,\; x\_,\; x\_,\; \backslash ldots)\backslash text$$ A sequence $x\_$ is eventually in $S$ if some tail of $x\_$ satisfies $x\_\; \backslash subseteq\; S.$
Let $\backslash tau$ 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 $X$ and $x\_$ a sequence therein. The sequence $x\_$ converges to a point $x\; \backslash in\; X,$ written $x\_\backslash overset\; x$ (when context allows, $x\_\backslash bull\backslash to\; x$), if, for every neighborhood $U\backslash in\backslash tau$ of $x,$ eventually $x\_$ is in $U.$ $x$ is then called a limit point of $x\_.$
A function $f\; :\; X\; \backslash to\; Y$ between topological spaces is sequentially continuous if $x\_\backslash bull\backslash to\; x$ implies $f(x\_\backslash bull)\backslash to\; f(x).$
Sequential closure/interior

Let $(X,\; \backslash tau)$ be a topological space and let $S\; \backslash subseteq\; X$ be a subset. The topological closure (resp. topological interior) of $S$ in $(X,\; \backslash tau)$ is denoted by $\backslash operatorname\_X\; S$ (resp. $\backslash operatorname\_X\; S$). The sequential closure of $S$ in $(X,\; \backslash tau)$ is the set$$\backslash operatorname(S)\; =\; \backslash left\backslash $$which defines a map, the sequential closure operator, on the power set of $X.$ If necessary for clarity, this set may also be written $\backslash operatorname\_(S)$ or $\backslash operatorname\_(S).$ It is always the case that $\backslash operatorname\_X\; S\; \backslash subseteq\; \backslash operatorname\_X\; S,$ but the reverse may fail. The sequential interior of $S$ in $(X,\; \backslash tau)$ is the set$$\backslash operatorname(S)\; =\; \backslash $$(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 $R,\; S\; \backslash subseteq\; X,$- $\backslash operatorname\_X(X\backslash setminus\; S)=X\backslash setminus\backslash operatorname\_X(S)$ and $\backslash operatorname\_X(X\backslash setminus\; S)=X\backslash setminus\backslash operatorname\_X(S)$;
- $\backslash operatorname(\backslash emptyset)\; =\; \backslash emptyset$ and $\backslash operatorname(\backslash emptyset)=\backslash emptyset$;
- $\backslash operatorname(S)\backslash subseteq\; S\backslash subseteq\backslash operatorname(S)$;
- $\backslash operatorname(R\backslash cup\; S)=\backslash operatorname(R)\backslash cup\backslash operatorname(S)$; and
- $\backslash operatorname(S)\backslash subseteq\backslash operatorname(\backslash operatorname(S)).$

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 $S$ is sequentially closed if $S=\backslash operatorname(S)$; equivalently, for all $s\_\backslash subseteq\; S$ and $x\; \backslash in\; X$ such that $s\_\backslash oversetx,$ we must have $x\backslash in\; S.$You cannot simultaneously apply this "test" to infinitely many subsets (for example, you can not use something akin to theaxiom 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 $S$ can be determined without it ever being necessary to consider any set other than $S.$
A set $S$ 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:
- $S\; =\; \backslash operatorname(S)$ or
- For all $x\_\backslash subseteq\; X$ and $s\; \backslash in\; S$ such that $x\_\backslash oversets,$ eventually $x\_$ is in $S$ (that is, there exists some integer $i$ such that the tail $x\_\; \backslash subseteq\; S$).

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 $\backslash alpha+1,$ define (as usual)$$(\backslash operatorname)^(S)=\backslash operatorname((\backslash operatorname)^\backslash alpha(S))$$and, for a limit ordinal $\backslash alpha,$ define$$(\backslash operatorname)^\backslash alpha(S)=\backslash bigcup\_\backslash text$$This process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index $\backslash omega\_1$ (thefirst 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 $X$ is the minimal ordinal at which, for any choice of $S,$ the above sequence will stabilize.
The transfinite sequential closure of $S$ is the terminal set in the above sequence: $(\backslash operatorname)^(S).$ The operator $(\backslash operatorname)^$ 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 $(X,\; \backslash tau)$ is sequential if it satisfies any of the following equivalent conditions:- $\backslash tau$ is its own sequential coreflection.
- Every sequentially open subset of $X$ is open.
- Every sequentially closed subset of $X$ is closed.
- For any subset $S\; \backslash subseteq\; X$ that is closed in $X,$ there exists someA Fréchet–Urysohn space is defined by the analogous condition for all such $x$:
For any subset $S\; \backslash subseteq\; X$ that is not closed in $X,$ ''for any'' $x\; \backslash in\; \backslash operatorname\_X(S)\; \backslash setminus\; S,$ there exists a sequence in $S$ that converges to $x.$

$x\backslash in\backslash operatorname(S)\backslash setminus\; S$ and a sequence in $S$ that converges to $x.$ Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12 - (Universal Property) For every topological space $Y,$ a map $f\; :\; X\; \backslash to\; Y$ is continuous if and only if it is sequentially continuous (if $x\_\; \backslash to\; x$ then $f\backslash left(x\_\backslash right)\; \backslash to\; f(x)$).
- $X$ is the quotient of a first-countable space.
- $X$ is the quotient of a metric space.

- 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 $X$ is sequentially closed (resp. open).
- $\backslash operatorname$ or $\backslash operatorname$ are idempotent.
- $\backslash operatorname(S)=\backslash bigcap\_$ or $\backslash operatorname(S)=\backslash bigcup\_$
- Any sequential neighborhood of $x\; \backslash in\; X$ can be shrunk to a sequentially-open set that contains $x$; formally, sequentially-open neighborhoods are a neighborhood basis for the sequential neighborhoods.
- For any $x\; \backslash in\; X$ and any sequential neighborhood $N$ of $x,$ there exists a sequential neighborhood $M$ of $x$ such that, for every $m\; \backslash in\; M,$ the set $N$ is a sequential neighborhood of $m.$

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 $N$-sequential. There exist topological vector spaces that are sequential but $N$-sequential (and thus not -sequential).
Fréchet–Urysohn spaces

A topological space $(X,\; \backslash tau)$ is called Fréchet–Urysohn if it satisfies any of the following equivalent conditions:- $X$ is hereditarily sequential; that is, every topological subspace is sequential.
- For every subset $S\; \backslash subseteq\; X,$ $\backslash operatorname\_X\; S\; =\; \backslash operatorname\_X\; S.$
- For any subset $S\; \backslash subseteq\; X$ that is not closed in $X$ and every $x\; \backslash in\; \backslash left(\backslash operatorname\_X\; S\backslash right)\; \backslash setminus\; S,$ there exists a sequence in $S$ that converges to $x.$

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 TExamples 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 $\backslash R$ and identify the set $\backslash Z$ of integers to a point. As a quotient of a metric space, the result is sequential, but it is not first countable. Everyfirst-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 $\backslash mathcal$ be a set of maps from Fréchet–Urysohn spaces to $X.$ Then the final topology that $\backslash mathcal$ induces on $X$ 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 $\backslash mathcal\backslash left(\backslash R^n\backslash right)$and the space $C^(U)$ 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 $C\_c^k(U)$ denote the space of $k$ -smooth test functions with its canonical topology and let $\backslash mathcal\text{'}(U)$ denote the space of distributions, the strong dual space of $C\_c^(U)$; neither are sequential (nor even an Ascoli space). On the other hand, both $C\_c^(U)$ and $\backslash mathcal\text{'}(U)$ are Montel spaces and, in thedual 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 $f\; :\; X\; \backslash to\; Y$ is a continuous opensurjection
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 $\backslash \backslash subseteq\; Y$ 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 $f\; :\; X\; \backslash to\; Y$ is a surjective map (not necessarily continuous) onto a Hausdorff sequential space $Y$ and $\backslash mathcal$ bases for the topology on $X,$ then $f\; :\; X\; \backslash to\; Y$ is an open map if and only if, for every $x\; \backslash in\; X,$ basic neighborhood $B\; \backslash in\; \backslash mathcal$ of $x,$ and sequence $y\_\; =\; \backslash left(y\_i\backslash right)\_^\; \backslash to\; f(x)$ in $Y,$ there is a subsequence of $y\_\backslash bull$ 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 thecategory 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