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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 if a convergent sequence is contained in a closed set then the limit of that sequence must be contained in 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

The related concepts of Fréchet–Urysohn spaces, T-sequential spaces, and -sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.

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

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.

Let be a set and let be a sequence in ; that is, a family of elements of , indexed by the natural numbers. In this article, means that each element in the sequence is an element of and, if is a map, then For any index the tail of starting at is the sequence A sequence is eventually in if some tail of satisfies

Let be a topology on and a sequence therein. The sequence converges to a point written (when context allows, ), if, for every neighborhood of eventually is in is then called a limit point of

A function between topological spaces is sequentially continuous if implies

Let be a topological space and let be a subset. The topological closure (resp. topological interior) of in is denoted by (resp. ).

The **sequential closure** of in is the set which defines a map, the **sequential closure operator**, on the power set of If necessary for clarity, this set may also be written or It is always the case that but the reverse may fail.

The **sequential interior** of in is the set (the topological space again indicated with a subscript if necessary).

Sequential closure and interior satisfy many of the nice properties of *topological* closure and interior: for all subsets

- and ;
Proof Fix If then there exists with But by the definition of sequential interior, eventually is in contradicting

Conversely, suppose ; then there exists a sequence with that is not eventually in By passing to the subsequence of elements not in we may assume that But then▮ - and ;
- ;
- ; and

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.

A set is sequentially closed if ; equivalently, for all and such that we must have ^{[note 1]}

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

- or
- For all and such that eventually is in (that is, there exists some integer such that the tail ).

A set is a **sequential neighborhood** of a point if it contains in its sequential interior; sequential neighborhoods need *not* be sequentially open (see § T- and N-sequential spaces below).

It is possible for a subset of to be sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.

As discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology. One can obtain an idempotent sequential closure via transfinite iteration: for a successor ordinal define (as usual) and, for a limit ordinal define This process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index (the first uncountable ordinal). Conversely, the **sequential order** of is the minimal ordinal at which, for any choice of the above sequence will stabilize.^{[2]}

The **transfinite sequential closure** of is the terminal set in the above sequence: The operator is idempotent and thus a closure operator. In particular, it defines a topology, the sequential coreflection. In the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).^{[3]}

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

- is its own sequential coreflection.
^{[4]} - Every sequentially open subset of is open.
- Every sequentially closed subset of is closed.
- For any subset that is
*not*closed in there exists some^{[note 2]}and a sequence in that converges to^{[5]} - (Universal Property) For every topological space a map is continuous if and only if it is sequentially continuous (if then ).
^{[6]} - is the quotient of a first-countable space.
- is the quotient of a metric space.

By taking and to be the identity map on in the universal property, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences. If two topologies agree on convergent sequences, then they necessarily have the same sequential coreflection. Moreover, a function from is sequentially continuous if and only if it is continuous on the sequential coreflection (that is, when pre-composed with ).

A **T-sequential space** is a topological space with sequential order 1, which is equivalent to any of the following conditions:^{[1]}

- The sequential closure (or interior) of every subset of is sequentially closed (resp. open).
- or are idempotent.
- or
- Any sequential neighborhood of can be shrunk to a sequentially-open set that contains ; formally, sequentially-open neighborhoods are a neighborhood basis for the sequential neighborhoods.
- For any and any sequential neighborhood of there exists a sequential neighborhood of such that, for every the set is a sequential neighborhood of

Being a T-sequential space is incomparable with being a sequential space; there are sequential spaces that are not T-sequential and vice-versa. However, a topological space is called a ** -sequential** (or **neighborhood-sequential**) if it is both sequential and T-sequential. An equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.^{[1]}

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

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

- is hereditarily sequential; that is, every topological subspace is sequential.
- For every subset
- For any subset that is not closed in and every there exists a sequence in that converges to

Fréchet–Urysohn spaces are also sometimes said to be "Fréchet," but should be confused with neither Fréchet spaces in functional analysis nor the T_{1} condition.

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.^{[7]}

Take the real line and identify the set of integers to a point. As a quotient of a metric space, the result is sequential, but it is not first countable.

Every first-countable space 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 be a set of maps from Fréchet–Urysohn spaces to Then the final topology that induces on is sequential.

A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences.^{[8]}^{[9]}

Schwartz space and the space of smooth functions, as discussed in the article on distributions, are both widely-used sequential spaces.^{[10]}^{[11]}

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

Arens' space is sequential, but not Fréchet–Urysohn.^{[12]}^{[13]}

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".)^{[14]}

Let denote the space of -smooth test functions with its canonical topology and let denote the space of distributions, the strong dual space of ; neither are sequential (nor even an Ascoli space).^{[10]}^{[11]} On the other hand, both and are Montel spaces^{[15]} 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).^{[10]}^{[16]}

Every sequential space has countable tightness and is compactly generated.

If is a continuous open surjection between two Hausdorff sequential spaces then the set of points with unique preimage is closed. (By continuity, so is its preimage in the set of all points on which is injective.)

If is a surjective map (not necessarily continuous) onto a Hausdorff sequential space and bases for the topology on then is an open map if and only if, for every basic neighborhood of and sequence in there is a subsequence of that is eventually in

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
^{[disputed – discuss]} - 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 (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".^{[17]}

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.

- Axiom of countability – property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not probably exist.
- Closed graph property – Graph of a map closed in the product space
- First-countable space – Topological space where each point has a countable neighbourhood basis
- Fréchet–Urysohn space – Property of topological space
- Sequence covering map

**^**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 can be determined without it ever being necessary to consider any set other than**^**A Fréchet–Urysohn space is defined by the analogous condition for all such :For any subset that is not closed in

*for any*there exists a sequence in that converges to

- ^
^{a}^{b}^{c}^{d}Snipes, Ray (1972). "T-sequential topological spaces" (PDF).*Fundamenta Mathematicae*.**77**(2): 95–98. doi:10.4064/fm-77-2-95-98. ISSN 0016-2736. **^***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.**^**Baron, S. (October 1968). "The Coreflective Subcategory of Sequential Spaces".*Canadian Mathematical Bulletin*.**11**(4): 603–604. doi:10.4153/CMB-1968-074-4. ISSN 0008-4395. S2CID 124685527.**^**"Topology of sequentially open sets is sequential?".*Mathematics Stack Exchange*.**^**Arkhangel'skii, A.V. and Pontryagin L.S., General Topology I, definition 9 p.12**^**Baron, S.; Leader, Solomon (1966). "Solution to Problem #5299".*The American Mathematical Monthly*.**73**(6): 677–678. doi:10.2307/2314834. ISSN 0002-9890. JSTOR 2314834.**^**"On sequential properties of Noetherian topological spaces" (PDF). 2004. Retrieved 30 Jul 2023.**^**Wilansky 2013, p. 224.**^**Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507- ^
^{a}^{b}^{c}Gabrielyan, Saak (2019). "Topological properties of strict -spaces and strong duals of Montel strict -spaces".*Monatshefte für Mathematik*.**189**(1): 91–99. arXiv:1702.07867. doi:10.1007/s00605-018-1223-6. - ^
^{a}^{b}T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36. **^**Engelking 1989, Example 1.6.19**^**Ma, Dan (19 August 2010). "A note about the Arens' space". Retrieved 1 August 2013.**^**math; Sleziak, Martin (Dec 6, 2016). "Example of different topologies with same convergent sequences".*Mathematics Stack Exchange*. StackOverflow. Retrieved 2022-06-27.**^**"Topological vector space".*Encyclopedia of Mathematics*. Retrieved September 6, 2020.It is a Montel space, hence paracompact, and so normal.

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