In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability.

- History
- Definitions
- Preliminaries
- Sequential closure/interior
- Sequentially open/closed sets
- Sequential spaces
- T-sequential and N {\displaystyle N} -sequential spaces
- Fréchet–Urysohn spaces
- Topology of sequentially open sets
- Properties of the topology of sequentially open sets
- Sequential continuity
- Sufficient conditions
- Examples
- Sequential spaces that are not Fréchet–Urysohn spaces
- Examples of non-sequential spaces
- Properties
- Categorical properties
- See also
- Notes
- Citations
- References

In any topological space every open subset has the following property: if a sequence in converges to some point in then the sequence will *eventually* be entirely in (i.e. there exists an integer such that all belong to ); Any set with this property is said to be **sequentially open**, regardless of whether or not it is open in However, it is possible for there to exist a subset that has this property but *fails* to be an open subset of Sequential spaces are *exactly* those topological spaces where a subset with this property never fails to be open. Sequential spaces can be viewed as exactly those spaces where for any single given subset knowledge of which sequences in converge to which point(s) of (and which don't) is sufficient to determine whether or not is closed in ^{ [note 1] } Thus sequential spaces are those spaces for which sequences in can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in ; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence. In any space that is *not* sequential, there exists a subset for which this "test" gives a "false positive."^{ [note 2] }

Alternatively, a space being sequential means that its topology if "forgotten", can be completely reconstructed using only sequences if one has *all* possible information about the convergence (or non-convergence) of sequences in and ** nothing more**. However, like all topologies, any topology that cannot be described entirely in terms of sequences can nevertheless be described entirely in terms of nets (also known as Moore–Smith sequences) or alternatively, in terms of filters. All first-countable spaces, which includes metric spaces, are sequential spaces.

There are other classes of topological spaces, such as Fréchet–Urysohn spaces, T-sequential spaces, and -sequential spaces, that are also defined in terms of how a space's topology interacts with sequences. Their definitions differ from that of sequential spaces in only subtle (but important) ways and it is often (initially) surprising that a sequential space does *not* necessarily have the properties of a Fréchet–Urysohn, T-sequential, or -sequential space.

Sequential spaces and -sequential spaces were introduced by S. P. Franklin.^{ [1] }

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, who was investigating the question of "what are the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences?" Franklin arrived at the definition above by noting that every first-countable space can be specified completely by the knowledge of its convergent sequences, and then he abstracted properties of first countable spaces that allowed this to be true.

Let be a set and let be a sequence in where a ** sequence in a set ** is by definition just a map from the natural numbers into If is a set then means that the is a sequence in If is a map then denotes the sequence Because the sequence is just a function this is consistent with the definition of function composition, meaning that

For any index the ** tail of starting at ** is the

The set of all tails of is denoted by

and it forms a *filter base* (also called a *prefilter*) on which is why it is called the ** prefilter of tails** or the

If is a subset then a sequence in is ** eventually in ** if there exists some index such that (that is, for any integer such that ).

Let be a topological space (*not* necessarily Hausdorff) and let be a sequence in The sequence ** converges ** in to a point written in and is called a

A point is called a ** cluster point ** or

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:

where or may be written if clarity is needed. The inclusion always holds but in general, set equality may *fail* to hold. The ** sequential closure operator** is the map defined by where denotes the power set of

The ** sequential interior** of in is the set:

where or may be written if clarity is needed.

As with the topological closure operator, and always hold and for all subsets

so that consequently,

However, it is in general possible that which in particular would imply that because the topological closure operator is idempotent, meaning that for all subsets

- Transfinite sequential closure

The ** transfinite sequential closure** is defined as follows: define to be define to be and for a limit ordinal define to be Then there is a smallest ordinal such that and for this is called the transfinite sequential closure of In fact, always holds where is the first uncountable ordinal. The transfinite sequential closure of is sequentially closed. Taking the transfinite sequential closure solves the idempotency problem above. The smallest such that for each is called

Let be a topological space (*not* necessarily Hausdorff) and let be a subset. It is known that the subset is open in if and only if whenever is a net in that converges in to a point then is eventually in where "** eventually in **" means that there exists some index such that for all satisfying The definition of a

The set is called ** sequentially open** if it satisfies any of the following equivalent conditions:

- Definition: Whenever a sequence in converges to some point of then that sequence is eventually in
- If is a sequence in and if there exists some is such that in then is eventually in (that is, there exists some integer such that the tail ).
- The set is sequentially closed in

The set is called ** sequentially closed** if it satisfies any of the following equivalent conditions:

- Definition: Whenever a sequence in converges in to some point then
- If is a sequence in and if there exists some is such that in then
- The set is sequentially open in

The complement of a sequentially open set is a sequentially closed set, and vice versa.

The set is called a ** sequential neighborhood** of a point if it satisfies any of the following equivalent conditions:

- Definition:
- Importantly, " is a sequential neighborhood of " is
*not*defined as: "there exists a sequentially open set such that "

- Importantly, " is a sequential neighborhood of " is
- Any sequence in that converges to is eventually in

Let

denote the set of all sequentially open subsets of where this may be denoted by is the topology is understood. Every open (resp. closed) subset of is sequentially open (resp. sequentially closed), which implies that

It is possible for the containment to be *proper*, meaning that there may exist a subset of that is sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.

A topological space is called a ** sequential space** if it satisfies any of the following equivalent conditions:

- Definition: Every sequentially open subset of is open.
- Every sequentially closed subset of is closed.
- For any subset that is
*not*closed insome for which there exists a sequence in that converges to*there exists*^{ [3] }- Contrast this condition to the following characterization of a Fréchet–Urysohn space:

- For any subset that is
*not*closed in andthere exists a sequence in that converges to*for every*

- This makes it obvious that every Fréchet–Urysohn space is a sequential space.

- is the quotient of a first countable space.
- is the quotient of a metric space.
: For every topological space a map is continuous if and only if it is sequentially continuous.*Universal property of sequential spaces*- A map is called
if for every and every sequence in if in then in This condition is equivalent to the map being continuous.*sequentially continuous* - Every continuous map is necessarily sequentially continuous but in general, the converse may fail to hold.

- A map is called

By taking and to be the identity map on in the last condition, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences.

Proof of the equivalences |
---|

Conversely, suppose for a contradiction that a subset is sequentially closed but not closed. By 3, there exists a sequence in that converges to a point in i.e. the limit lies outside This contradicts sequential closedness of |

A sequential space may fail to be a T-sequential space and also a T-sequential space may fail to be a sequential space. In particular, it should *not* be assumed that a sequential space has the properties described in the next definitions.

A topological space is called a ** T-sequential space** (or

- Definition: The sequential interior of every subset of is sequentially open.
- The sequential closure of every subset of is sequentially closed.
- For all
- The inclusion always holds for every

- For all
- The inclusion always holds for all

- For all is equal to the union of all subsets of that are sequentially open in
- For all is equal to the intersection of all subsets of that contain and are sequentially closed in
- For all the collection of all sequentially open neighborhoods of in forms a neighborhood basis at for the set of all sequential neighborhoods of
- This means for any and any sequential neighborhood of there exists a sequentially open set such that
- Here, the exact definition of "sequential neighborhood" is important because recall that " is a sequential neighborhood of " was defined to mean that

- For any and any sequential neighborhood of there exists a sequential neighborhood of such that for every the set is a sequential neighborhood of

As with T-sequential spaces, it should *not* be assumed that a sequential space has the properties described in the next definition.

A topological space is called an ** -sequential** (or

- Definition: For every if a set is a
*sequential*neighborhood of then is a neighborhood of in- Recall that being a sequential neighborhood (resp. a neighborhood) of means that (resp. ).

- is both sequential and T-sequential.

Every first-countable space is -sequential.^{ [1] } There exist topological vector spaces that are sequential but *not*-sequential (and thus not T-sequential).^{ [1] } where recall that every metrizable space is first countable. There also exist topological vector spaces that are T-sequential but not sequential.^{ [1] }

Every Fréchet–Urysohn space is a sequential space but there exist sequential spaces that are not Fréchet–Urysohn.^{ [4] }^{ [5] } Consequently, it should *not* be assumed that a sequential space has the properties described in the next definition.

A topological space is called ** Fréchet–Urysohn space ** if it satisfies any of the following equivalent conditions:

- Definition: For every subset
- Every topological subspace of is a sequential space.
- For any subset that is
*not*closed in and*for every*there exists a sequence in that converges to

Fréchet–Urysohn spaces are also sometimes said to be ** Fréchet**, which should not be confused with Fréchet spaces in functional analysis; confusingly,

Let denote the set of all sequentially open subsets of the topological space Then is a topology on that contains the original topology that is,

Proofs |
---|

Let be sequentially open. It is now shown that its complement is sequentially closed; that is, that a convergent sequence of elements of has its limit in Suppose for a contradiction that then there exists some integer such that which contradicts the fact that all are supposed to be in The converse will now be shown; that is, it is now shown that if is sequentially closed then its complement is sequentially open. Let be a sequence in such that and suppose for a contradiction that for any i.e. for all integers there exists Define by recursion the subsequence of elements of : set and then that is, and It is convergent as a subsequence of a convergent sequence, and all its elements are in Hence the limit has to be in which contradicts that The sequence is therefore eventually in It is now shown that the set of sequentially open subsets is a topology. Specifically, this means that and are sequentially open, arbitrary unions of sequentially open subset is sequentially open and finite intersections of sequentially open subsets is sequentially open. Any empty sequence satisfies any property and any sequence in is eventually in Let be a family of sequentially open subsets, let and let be a sequence in converging to being in the union means there exists such that and by sequential openness, the sequence is eventually in Finally, if is a finite intersection of sequentially open subsets, then a sequence converging to eventually converges to each of the i.e. for all satisfying there exists some such that Taking one has The generated sequential topology is finer than the original one, meaning that if is open, then it is sequentially open. Let be a sequence in converging to Since is open, it is a neighborhood of and by definition of convergence, there exists such that |

The topological space is said to be * sequentially Hausdorff* if is a Hausdorff space.

Every sequential space has countable tightness.

The topological space is always a sequential space (even if is not),^{ [6] } and has the same convergent sequences and limits as Explicitly, this means that if and is a sequence in then in if and only if in

If is any topology on such that for every and every sequence in

then necessarily

If is continuous then so is

A map is called ** sequentially continuous ** if for every sequence in and every if in then necessarily in which happens if and only if

is continuous.

Every continuous map is sequentially continuous although in general, the converse may fail to hold. In fact, a space is a sequential space if and only if it has the following * universal property for sequential spaces*:

- For every topological space and every map the map is continuous if and only if it is sequentially continuous.

Every first-countable space is sequential, hence each second-countable space, metric space, and discrete space is sequential. Every first-countable space is a Fréchet–Urysohn space and every Fréchet-Urysohn space is sequential. Thus every metrizable and pseudometrizable space is a sequential space and a Fréchet–Urysohn space.

A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences.^{ [7] }^{ [8] }

Let be a set and let be a family of -valued maps with each map being of the form where the domain is some topological space. If every domain is a Fréchet–Urysohn space then the final topology on induced by makes into a sequential space.

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.

- Sequential spaces that are not first countable

Take the real line and identify the set of integers to a point. It is a sequential space since it is a quotient of a metric space. But it is not first countable.

The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces. Let denote the Schwartz space and let denote the space of smooth functions on an open subset where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. Both and as well as the strong dual spaces of both these of spaces, are complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also paracompact ^{ [9] } normal reflexive barrelled spaces. The strong dual spaces of both and are sequential spaces but *neither one* of these duals is a Fréchet-Urysohn space.^{ [10] }^{ [11] }

Every infinite-dimensional Montel DF-space is a sequential space but *not* a Fréchet–Urysohn space.

- Spaces of test functions and distributions

Let denote the space of test functions with its canonical LF topology, which makes it into a distinguished strict LF-space and let denote the space of distributions, which by definition is the strong dual space of These two space, which completely underpin the theory of distributions and which have many nice properties, are nevertheless prominent examples of spaces that are *not* sequential spaces (and thus neither Fréchet–Urysohn spaces nor -sequential spaces).

Both and are complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also paracompact ^{ [9] } normal reflexive barrelled spaces. It is known that 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 (i.e. pointwise),^{ [12] } which in particular, is the reason why a sequence of distributions converges in (with is given strong dual topology) if and only if it converges pointwise. The space is also a Schwartz topological vector space. Nevertheless, neither nor its strong dual is a sequential space (not even an Ascoli space).^{ [10] }^{ [11] }

- Cocountable topology

Another example of a 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. In fact, one could say that the cocountable topology on an uncountable set is "sequentially discrete".

If is a continuous open surjection between two Hausdorff sequential spaces then the set is a closed subset of the set is a closed subset of that satisfies and the restriction is injective.

If is a surjective map (not assumed to be continuous) onto a Hausdorff sequential space and if is a basis for the topology on then is an open map if and only if for every and every basic neighborhood of if in then necessarily Here, denotes the image (or range) of the sequence/map

The full subcategory **Seq** of all sequential spaces is closed under the following operations in the category **Top** of topological spaces:

- Quotients
- Continuous closed or open images
- Sums
- Inductive limits
- Open and closed subspaces

The category **Seq** is *not* closed under the following operations in **Top**:

- Continuous images
- Subspaces
- Finite products

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 (i.e., 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".

- Axioms of countability
- Closed graph – Graph of a function that is also a closed subset of the product space
- First-countable space – Topological space where each point has a countable neighbourhood basis
- Fréchet–Urysohn space
- Sequence covering map

- ↑ This interpretation assumes that you make this determination
*only*to the given set and not to other sets; said differently, you cannot simultaneously apply this "test" to infinitely many subsets (e.g. you can not use something akin to the axiom of choice). It is in Fréchet-Urysohn spaces that the closure of a set can be determined without it ever being necessary to consider any set other than There exist sequential spaces that are*not*Fréchet-Urysohn spaces. - ↑ Although this "test" (which attempts to answer "is this set open (resp. closed)?") could potentially give a "false positive," it can never give a "false negative;" this is because every open (resp. closed) subset is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set that really is open (resp. closed).

- 1 2 3 4 5 6 Snipes, Ray F. "T-sequential topological spaces"
- ↑
- Arhangel'skiĭ, A. V.; Franklin, S. P. (1968). "Ordinal invariants for topological spaces".
*Michigan Math. J*.**15**(3): 313–320. doi: 10.1307/mmj/1029000034 .

- Arhangel'skiĭ, A. V.; Franklin, S. P. (1968). "Ordinal invariants for topological spaces".
- ↑ Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12
- ↑ Engelking 1989, Example 1.6.18
- ↑ Ma, Dan (19 August 2010). "A note about the Arens' space" . Retrieved 1 August 2013.
- ↑ "Topology of sequentially open sets is sequential?".
- ↑ Wilansky 2013, p. 224.
- ↑ Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507
- 1 2 "Topological vector space".
*Encyclopedia of Mathematics*. Encyclopedia of Mathematics. Retrieved September 6, 2020.It is a Montel space, hence paracompact, and so normal.

- 1 2 Gabriyelyan, Saak "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces" (2017)
- 1 2 T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.
- ↑ Trèves 2006, pp. 351-359.

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

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 of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, **open sets** are a generalization of open intervals in the real line. In a metric space—that is, when a distance function is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, the **closure** of a subset *S* of points in a topological space consists of all points in *S* together with all limit points of *S*. The closure of *S* may equivalently be defined as the union of *S* and its boundary, and also as the intersection of all closed sets containing *S*. Intuitively, the closure can be thought of as all the points that are either in *S* or "near" *S*. A point which is in the closure of *S* is a point of closure of *S*. The notion of closure is in many ways dual to the notion of interior.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

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 closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, a **limit point** of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences. A **cluster point** or **accumulation point** of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is *not* synonymous with a "cluster/accumulation point"; by definition, the similarly named notion of a limit point of a filter refers to a point that the filter converges to.

In mathematics, more specifically in topology, an **open map** is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a **closed map** is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

In mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

In general topology and related areas of mathematics, the **final topology** on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent **characterizations of the category of topological spaces**. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.

In the field of topology, a **Fréchet–Urysohn space** is a topological space with the property that for every subset the closure of in is identical to the *sequential* closure of in Fréchet–Urysohn spaces are a special type of sequential space.

In topology, a subfield of mathematics, *filters* are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called *ultrafilters* have many useful technical properties and they may often be used in place of arbitrary filters.

In functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, including those that are not metrizable or Hausdorff.

In mathematics, an ** LB-space**, also written

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

In mathematics, particularly in functional analysis and topology, the **closed graph theorem** is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

In mathematics, specifically topology, a **sequence covering map** is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include *sequentially quotient* maps, *sequence coverings*, *1-sequence coverings*, and *2-sequence coverings*. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.

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