In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.
In any topological space (X, τ), every open subset S has the following property: if a sequence x_{•} = (x_{i})^{∞}
_{i=1} in X converges to some point in S then the sequence will eventually be entirely in S (i.e. there exists an integer N such that x_{N}, x_{N+1} , ... all belong to S).
(Any set with this property is said to be sequentially open, regardless of whether or not it is open in (X, τ)).
However, it is possible for there to exist a subset S that has this property but fails to be an open subset of X.
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 X where for any single given subset S ⊆ X, knowledge of which sequences in X converge to which point(s) of X (and which don't) is sufficient to determine whether or not S is closed in X.^{[note 1]}
Thus sequential spaces are those spaces X for which sequences in X can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in X; 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 (X, τ) being sequential means that its topology τ, if "forgotten", can be completely reconstructed using only sequences if one has all possible information about the convergence (or nonconvergence) of sequences in (X, τ) 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 firstcountable spaces, which includes metric spaces, are sequential spaces.
There are other classes of topological spaces, such as Fréchet–Urysohn spaces, Tsequential spaces, and Nsequential 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) suprising that a sequential space does not necessarily have the properties of a Fréchet–Urysohn, Tsequential, or Nsequential space.
Sequential spaces and Nsequential spaces were introduced by S. P. Franklin.^{[1]}
History
Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is originally due to S. P. Franklin (aka Stan 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 firstcountable 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.
Definitions
Preliminaries
Let X be a set and let x_{•} = (x_{i})^{∞}
_{i=1} be a sequence in X, where recall that a sequence in a set X is by definition just a map from the natural numbers ℕ into X.Notation and definition: For any index i, the tail of x_{•} starting at i is the set:
 x_{≥ i} := { x_{j} : j ≥ i } := { x_{i}, x_{i+1}, x_{i+2}, ... }.
The set Tails(x_{•}) := { x_{≥ i} : i ∈ ℕ } of all tails of x_{•} is called the sequential filter base of tails of x_{•} and it defines a filter base on X.
Definition: If S ⊆ X is a subset then we say that a sequence x_{•} is eventually in S if there exists some index i such that x_{≥ i} ⊆ S (that is, x_{j} ∈ S for any integer j such that j ≥ i ).
Notation: If f : X → Y is a map then f (x_{•}) := (���f (x_{i}))^{∞}
_{i=1} denotes the sequence:
 f (x_{1}), f (x_{2}), f (x_{3}), ...
(note that since the sequence x_{•} is just a function x_{•} : ℕ → X, this is consistent with the definition of function composition; i.e. f (x_{•}) := f ∘ x_{•} ).
Let (X, τ) be a topological space (not necessarily Hausdorff) and let x_{•} = (x_{i})^{∞}
_{i=1} be a sequence in X.Notation and definition: We say that x_{•} converges in (X, τ) to a point x ∈ X, written x_{•} → x in (X, τ), and we call x a limit point of x_{•} if for every neighborhood U of x in (X, τ), x_{•} is eventually in U.
Notation: As usual, if we write lim x_{•} = x then we mean that x_{•} → x in (X, τ) and x is the only limit point of x_{•} in (X, τ) (i.e. if x_{•} → z in (X, τ) then z = x).
 Note that if (X, τ) is not Hausdorff then it's possible for a sequence to converge to two or more distinct points.
Definition: We say that x ∈ X is a cluster point or accumulation point of x_{•} in (X, τ) if for every neighborhood U of x in (X, τ) and every i ∈ ℕ, there exists some integer j ≥ i such that x_{j} ∈ U (or said differently, if and only if for every neighborhood U of x and every i ∈ ℕ, we have U ∩ x_{≥ i} ≠ ∅).
Sequential closure/interior
Definition and notation: Let (X, τ) be a topological space and let S ⊆ X be a subset.
 The sequential closure of S in (X, τ) is the set:
where we may write SeqCl_{X} S or SeqCl_{(X, τ)} S if clarity is needed.
 SeqCl S := [ S ]_{seq} := { x ∈ X : there exists a sequence s_{•} = (s_{i})^{∞}
_{i=1} in S such that s_{•} → x in (X, τ) } The sequential closure operator is the map induced by SeqCl. That is, it is the map SeqCl : ℘(X) → ℘(X) defined by S ↦ SeqCl S, where ℘(X) denotes the power set of X.
 The sequential interior of S in (X, τ) is the set:
where we may write SeqInt_{X} S or SeqInt_{(X, τ)} S if clarity is needed.
 SeqInt S := { s ∈ S : whenever x_{•} = (x_{i})^{∞}
_{i=1} is a sequence in X such that x_{•} → s in (X, τ), then x_{•} is eventually in S } = { s ∈ S : there does not exist a sequence x_{•} = (x_{i})^{∞}
_{i=1} in X ∖ S such that x_{•} → s in (X, τ) } = X ∖ SeqCl ( X ∖ S )
 We denote the topological closure (resp. topological interior) of S in X by Cl_{X} S (resp. Int_{X} S ).
 For any subsets R and S of X, we have:
 SeqCl ∅ = ∅
 SeqCl (R ∪ S) = (SeqCl R) ∪ (SeqCl S)
 S ⊆ SeqCl S
 SeqCl S ⊆ SeqCl ( SeqCl S )
 In particular, it's possible that SeqCl ( SeqCl S ) ≠ SeqCl S, which of course would imply that SeqCl S ≠ Cl S (since recall that the topological closure operator is idempotent, meaning that Cl (���Cl S ) = Cl S for all subsets S).
 Transfinite sequential closure
The transfinite sequential closure is defined as follows: define to be A, 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 A. (In fact, we always have , where is the first uncountable ordinal.) The transfinite sequential closure of A is sequentially closed. Taking the transfinite sequential closure solves the idempotency problem above.
The smallest such that for each is called sequential order of the space X.^{[2]} This ordinal invariant is welldefined for sequential spaces.
Sequentially open/closed sets
Definition: Let (X, τ) be a topological space (not necessarily Hausdorff) and let S ⊆ X be a subset.
 The set S is called sequentially open if it satisfies any of the following equivalent conditions:
 Whenever a sequence in X converges to some point of S, then that sequence is eventually in S.
 If x_{•} = (x_{i})^{∞}
_{i=1} is a sequence in X and if there exists some s ∈ S is such that x_{•} → s in (X, τ), then x_{•} is eventually in S (i.e. there exists some integer i such that x_{≥ i} ⊆ S ). S = SeqInt_{X} S.
 The complement X ∖ S is sequentially closed in (X, τ).
 The set S is called sequentially closed if it satisfies any of the following equivalent conditions:
 Whenever is a sequence in S converges in (X, τ) to some point x ∈ X, then x ∈ S.
 If s_{•} = (s_{i})^{∞}
_{i=1} is a sequence in S and if there exists some x ∈ X is such that s_{•} → x in (X, τ), then x ∈ S. S = SeqCl_{X} S.
 The complement X ∖ S is sequentially open in (X, τ).
 The set S is called a sequential neighborhood of a point x ∈ X if it satisfies any of the following equivalent conditions:
 x ∈ SeqInt S;
 Note that "S is a sequential neighborhood of x" is not defined as: "there exists a sequentially open set U such that x ∈ U ⊆ S."
 Any sequence in X that converges to x is eventually in S.
Notation: We denote the set of all sequentially open subsets of (X, τ) by SeqOpen(X, τ) or simply SeqOpen(X).
Note that:
 The complement of a sequentially open set is a sequentially closed set, and vice versa.
 Every open (resp. closed) subset of X is sequentially open (resp. sequentially closed), which implies that
 τ ⊆ SeqOpen(X, τ).
 It's possible for the containment τ ⊆ SeqOpen(X, τ) to be proper, meaning that there may exist a subset of X that is sequentially open but not open. Similarly, it's possible for there to exist a sequentially closed subset that is not closed.
Sequential spaces
Definition: A sequential space is a space (X, τ) satisfying any of the following equivalent conditions:
 Every sequentially open subset of X is open.
 Every sequentially closed subset of X is closed.
 For any subset S ⊆ X that is not closed in X, there exists some x ∈ (Cl S) ∖ S for which there exists a sequence in S that converges to x.^{[3]}
 Remark: Contrast this condition to the following characterization of a Fréchet–Urysohn space:
 For any subset S ⊆ X that is not closed in X and for every x ∈ (Cl S) ∖ S, there exists a sequence in S that converges to x.
 This makes it obvious that every Fréchet–Urysohn space is a sequential space.
 X is the quotient of a first countable space.
 X is the quotient of a metric space.
 For every topological space Y, a map f : X → Y is continuous if and only if it is sequentially continuous.
 A map f : X → Y is called sequentially continuous if for every x ∈ X and every sequence x_{•} = (x_{i})^{∞}
_{i=1} in X, if x_{•} → x in X then f (x_{•}) = ( f (x_{i}))^{∞}
_{i=1} → f (x) in Y. Every continuous map is necessarily sequentially continuous but in general, the conserve may fail to hold.
By taking Y := X and f to be the identity map on X 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


(1) ⇔ (2): Assume that any sequentially open subsets is open and let F be sequentially closed. It is proved above that the complement U = X ∖ F is sequentially open and thus open so that F is closed. The converse is similar. (2) ⇔ (3): Contraposition of 2 says that "S not closed implies S not sequentially closed", and hence there exists a sequence of elements of S that converges to a point outside of S. Since the limit is necessarily adherent to S, it is in the closure of S. Conversely, suppose for a contradiction that a subset S := F is sequentially closed but not closed. By 3, there exists a sequence in F that converges to a point in , i.e. the limit lies outside F. This contradicts sequential closedness of F. 
Tsequential and Nsequential spaces
A sequential space may fail to be a Tsequential space and also a Tsequential 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.
Definition: A topological space (X, τ) is called a Tsequential space if it satisfies one of the following equivalent conditions:^{[1]}
 The sequential interior of every subset of X is sequentially open.
 The sequential closure of every subset of X is sequentially closed.
 For all S ⊆ X, SeqCl ( SeqCl S ) = SeqCl S.
 Note that SeqCl S ⊆ SeqCl ( SeqCl S ) always holds for all S ⊆ X.
 For all S ⊆ X, SeqInt S = SeqInt (SeqInt S ).
 Note that SeqInt (SeqInt S ) ⊆ SeqInt S always holds for all S ⊆ X.
 For all S ⊆ X, SeqInt S is equal to the union of all subsets of S that are sequentially open in (X, τ).
 For all S ⊆ X, SeqCl S is equal to the intersection of all subsets of X that contain S and are sequentially closed in (X, τ).
 For all x ∈ X, the collection of all sequentially open neighborhoods of x in (X, τ) forms a neighborhood basis at x for the set of all sequential neighborhoods of x.
 This means for any x ∈ X and any sequential neighborhood N of x, there exists a sequentially open set U such that x ∈ U ⊆ N.
 Here the exact definition of "sequential neighborhood" is important because recall that "N is a sequential neighborhood of x" was defined to mean that x ∈ SeqInt N.
 For any x ∈ X and any sequential neighborhood N of x, there exists a sequential neighborhood M of x such that for every m ∈ M, the set N is a sequential neighborhood of m.
As with Tsequential spaces, it should not be assumed that a sequential space has the properties described in the next definition.
Definition: A topological space (X, τ) is called an Nsequential (or neighborhoodsequential) space if it satisfies any of the following equivalent conditions:^{[1]}
 For every x ∈ X, if a set N ⊆ X is a sequential neighborhood of x then N is a neighborhood of x in (X, τ).
 Recall that N being a sequential neighborhood (resp. neighborhood) of x means that x ∈ SeqInt N (resp. x ∈ Int N).
 X is both sequential and Tsequential.
Every firstcountable space is Nsequential.^{[1]} There exist topological vector spaces that are sequential but not Nsequential (and thus not Tsequential).^{[1]} where recall that every metrizable space is first countable. There also exist topological vector spaces that are Tsequential but not sequential.^{[1]}
Fréchet–Urysohn spaces
Every Fréchet–Urysohn space is a sequential space but there exist sequential spaces that are not Fréchet–Urysohn.^{[4]}^{[5]} Thus it should not be assumed that a sequential space has the properties described in the next definition.
Definition: We say that a topological space (X, τ) is a Fréchet–Urysohn space if it satisfies any of the following equivalent conditions:
 For every subset S ⊆ X, SeqCl_{X} S = Cl_{X} S;
 Every topological subspace of X is a sequential space;
 For any subset S ⊆ X that is not closed in X and for every x ∈ (Cl S) ∖ S, there exists a sequence in S that converges to x.
Note that Fréchet–Urysohn spaces are also sometimes said to be Fréchet, which should not be confused with Fréchet spaces in functional analysis; confusingly, Fréchet space in topology is also sometimes used as a synonym for T_{1} space.
Topology of sequentially open sets
Let denote the set of all sequentially open subsets of the topological space (X, τ). Then is a topology on X that contains the original topology (i.e. ).
Proofs


Let U be sequentially open. Let us show that its complement F = X ∖ U is sequentially closed, i.e. that a convergent sequence x_{•} = (x_{i})^{∞} Suppose for a contradiction that , then there exists some integer N > 0 such that , which contradicts the fact that all x_{n} are supposed to be in F. Conversely if F is sequentially closed, let us show that its complement U = X ∖ F is sequentially open. Let x_{•} = (x_{i})^{∞} Let us show that the set of sequentially open subsets is a topology, i.e. ∅ and X 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 X is eventually in X. Let (U_{i})_{i∈I} be a family of sequentially open subsets, let , and let x_{•} = (x_{i})^{∞} The generated sequential topology is finer than the original one, i.e. if U is open, then it is sequentially open. Let x_{•} = (x_{i})^{∞} 
Properties of the topology of sequentially open sets
 Every sequential space has countable tightness.

The space has the same convergent sequences and limits as (X, τ) (i.e. if x ∈ X and x_{•} = (x_{i})^{∞}
_{i=1} is a sequence in X, then in (X, τ) if and only if in ).  is a sequential space.^{[6]}
 If is any topology on X such that a sequence in X converges to a point of X in (X, τ) if and only if it does so in , then necessarily .
 If is continuous, then so is .
Sufficient conditions
Every firstcountable space is sequential, hence each secondcountable space, metric space, and discrete space is sequential. Every firstcountable space is a Fréchet–Urysohn space and every FréchetUrysohn 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]}
Examples
Every CWcomplex 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 ℝ and identify the set ℤ 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.
Sequential spaces that are not Fréchet–Urysohn spaces
The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces. Let 𝒮(ℝ^{n}) denote the Schwartz space and let denote the space of smooth functions on an open subset U ⊆ ℝ^{n}, where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. Both 𝒮(ℝ^{n}) 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 𝒮(ℝ^{n}) and are sequential spaces but neither one of these duals is a FréchetUrysohn space.^{[10]}^{[11]}
Every infinitedimensional Montel DFspace is a sequential space but not a Fréchet–Urysohn space.
Examples of nonsequential spaces
 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 LFspace 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 Nsequential 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".
Categorical properties
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, CWcomplexes, and differentiable manifolds and that is closed under colimits, quotients, and other "certain reasonable identities" that Norman Steenrod described as "convenient".
See also
 Axioms of countability
 Closed graph – A graph of a function that is also a closed subset of the product space
 Firstcountable space – A topological space where each point has a countable neighbourhood basis
 Fréchet–Urysohn space
Notes
 ^ This interpretation assumes that you make this determination only to the given set S and not to other sets; said differently, you cannot simultaneously apply this "test" to infinitely many subsets (e.g. you can't use something akin to the axiom of choice). It is in FréchetUrysohn spaces that the closure of a set S can be determined without it ever being necessary to consider any set other than S. There exist sequential spaces that are not FréchetUrysohn 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 S is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set S that really is open (resp. closed).
References
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Snipes, Ray F. "Tsequential 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.
 ^ Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12
 ^ Engelking 1989, Example 1.6.18
 ^ Ma, Dan. "A note about the Arens' space". Retrieved 1 August 2013.
 ^ https://math.stackexchange.com/questions/3737020/topologyofsequentiallyopensetsissequential
 ^ Wilansky 2013, p. 224.
 ^ Dudley, R. M., On sequential convergence  Transactions of the American Mathematical Society Vol 112, 1964, pp. 483507
 ^ ^{a} ^{b} "Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020.
It is a Montel space, hence paracompact, and so normal.
 ^ ^{a} ^{b} Gabriyelyan, Saak "Topological properties of Strict LFspaces and strong duals of Montel Strict LFspaces" (2017)
 ^ ^{a} ^{b} T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 3136.
 ^ Trèves 2006, pp. 351359.
Bibliography
 Arkhangel'skii, A.V. and Pontryagin, L.S., General Topology I, SpringerVerlag, New York (1990) ISBN 3540181784.
 Booth, P.I. and Tillotson, A., Monoidal closed, cartesian closed and convenient categories of topological spaces Pacific J. Math., 88 (1980) pp. 35–53.
 Engelking, R., General Topology, Heldermann, Berlin (1989). Revised and completed edition.
 Franklin, S. P., "Spaces in Which Sequences Suffice", Fund. Math. 57 (1965), 107115.
 Franklin, S. P., "Spaces in Which Sequences Suffice II", Fund. Math. 61 (1967), 5156.
 Goreham, Anthony, "Sequential Convergence in Topological Spaces"
 Steenrod, N.E., A convenient category of topological spaces, Michigan Math. J., 14 (1967), 133152.
 Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 9780486453521. OCLC 853623322.
 Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 9780486493534. OCLC 849801114.