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Sequential space

In topology, a sequential space is a topological space X in which every sequentially closed subset is closed, meaning that the closure of any subset A \subseteq X coincides with its sequential closure (the smallest sequentially closed set containing A).^[1] This property ensures that the topology of X is completely characterized by the convergent and divergent sequences within it, allowing sequential convergence to determine open and closed sets without needing more general nets or filters. Sequential spaces occupy an intermediate position in the hierarchy of convergence properties in general topology. Every first-countable space, such as metric spaces, is sequential, as sequences suffice to probe neighborhoods in such settings.^[1] More strongly, Fréchet-Urysohn spaces—where limits of sequences in a set can be expressed as limits of sequences from that set—are sequential, but the converse does not hold. An equivalent characterization is that a space is sequential if and only if it is the quotient of a metric space, which implies that sequential spaces inherit many sequential-like behaviors from metrizable prototypes.^[1] Notably, quotients of sequential spaces remain sequential, preserving this structure under continuous surjections.^[1] Examples of sequential spaces abound in classical topology. All metric spaces, including Euclidean spaces \mathbb{R}^n, are sequential due to their first-countability.^[1] CW-complexes, fundamental objects in algebraic topology, are also sequential as quotients of metrizable spaces like simplicial complexes.^[1] However, not all topological spaces are sequential; for instance, the cocountable topology on an uncountable set fails this property because certain non-closed sets lack witnessing sequences.^[2] The concept of sequential spaces is crucial for studying convergence and continuity in non-metrizable settings, bridging metric intuition with general topological phenomena. In sequential spaces, a function f: X \to Y is continuous if and only if it preserves sequential limits, simplifying proofs in areas like functional analysis and algebraic topology.^[3] This framework also aids in constructing counterexamples and exploring tightness properties, such as countable tightness, which sequential spaces possess.^[4]

Definitions and Basic Concepts

Definition of Sequential Space

In topology, a sequential space is defined as a topological space X in which every sequentially open set is open. A subset U \subseteq X is sequentially open if, whenever a sequence in X converges to a point x \in U, the sequence is eventually contained in U. This definition, introduced by S. P. Franklin, captures spaces where the topology can be fully determined by the behavior of sequences.^[5] Equivalent characterizations include the condition that the closure of any subset A \subseteq X coincides with its sequential closure, defined as the set of all limits of sequences in A. Additionally, in sequential spaces, a function f: X \to Y to another topological space Y is continuous if and only if it is sequentially continuous, meaning it preserves the convergence of sequences. These equivalences highlight how sequences alone suffice to delineate topological structure without recourse to more general constructs like nets or filters.^[5] The motivation for sequential spaces arises from the observation that, while sequences fully characterize convergence and continuity in familiar settings like metric spaces, they generally fail to do so in arbitrary topological spaces. Sequential spaces bridge this gap by ensuring sequences probe the topology effectively. Examples include all metric spaces and, more broadly, all first-countable spaces, where a local countable basis guarantees the existence of suitable sequences for limits.^[6]

Sequential Coreflection

In the category of topological spaces, the sequential coreflection is given by the coreflector functor S: \mathbf{Top} \to \mathbf{Seq}, where \mathbf{Seq} is the full subcategory of sequential spaces. For any topological space X, the sequential coreflection SX has the same underlying set as X and is equipped with the finest topology such that the convergent sequences in SX coincide exactly with those in X. This topology is finer than the original topology on X.^[7] The topology on SX is constructed by taking the collection of all sequentially open sets in X as the open sets. A subset U \subseteq X is sequentially open if, whenever a sequence in X converges to a point x \in U, the sequence is eventually contained in U. Every open set in X is sequentially open, but in general there are additional sequentially open sets when X is not sequential, making the topology on SX strictly finer unless X is already sequential. Equivalently, the closed sets in SX are precisely the sequentially closed subsets of X, where a subset C \subseteq X is sequentially closed if every sequence in C that converges in X has its limit in C.^[7] The universal property of the sequential coreflection states that for any sequential space Y and any continuous map f: X \to Y, there exists a unique continuous map g: SX \to Y such that g \circ \eta_X = f, where \eta_X: X \to SX is the identity map on underlying sets (the unit of the adjunction). Here, g = f (viewed as a map from SX to Y), and its continuity holds because both SX and Y are sequential, and f preserves the shared family of convergent sequences from X. The counit \varepsilon_X: SX \to X, also the identity on underlying sets, is continuous (as the topology on X is coarser) but not necessarily open. The unit \eta_X: X \to SX is not necessarily continuous.^[7]

Sequential Operators

Sequential Closure

In a topological space X, the sequential closure of a subset A \subseteq X, denoted \cl_{\seq}(A), is the set of all points x \in X that are limits of convergent sequences with terms in A. This operator captures the points reachable from A via sequential convergence, providing a sequence-based approximation to the topological structure.^[8] A point x belongs to \cl_{\seq}(A) if and only if there exists a sequence (x_n)_{n \in \mathbb{N}} in A such that x_n \to x in X. The sequential closure satisfies \cl_{\seq}(A) \subseteq \cl(A), where \cl(A) denotes the standard topological closure of A. In sequential spaces, this inclusion is an equality, meaning the topology is fully determined by sequential limits: \cl_{\seq}(A) = \cl(A) for every A \subseteq X.^[5]^[9] The sequential closure operator is monotonic, preserving inclusions: if A \subseteq B, then \cl_{\seq}(A) \subseteq \cl_{\seq}(B). However, it is not necessarily additive, as \cl_{\seq}(A \cup B) may strictly contain \cl_{\seq}(A) \cup \cl_{\seq}(B) in general topological spaces. These properties align with the Kuratowski closure axioms except for idempotence and additivity, distinguishing the sequential closure from the full topological closure operator while highlighting its role in sequence-determined topologies. Sequentially closed sets, where A = \cl_{\seq}(A), coincide with closed sets precisely in sequential spaces.^[9]

Sequential Interior

In a topological space X, the sequential interior of a subset A \subseteq X, denoted \operatorname{int}_{\text{seq}}(A), consists of all points x \in A such that no sequence in X \setminus A converges to x.^[10] This operator captures points in A that cannot be approached by sequences from outside A. Equivalently, \operatorname{int}_{\text{seq}}(A) = A \setminus \operatorname{cl}_{\text{seq}}(X \setminus A), where \operatorname{cl}_{\text{seq}} denotes the sequential closure operator, highlighting the duality between sequential interior and sequential closure.^[10]^[11] The sequential interior satisfies \operatorname{int}_{\text{seq}}(A) \subseteq \operatorname{int}(A), the standard interior, because sequentially open sets are open, so the largest sequentially open subset of A lies within the largest open subset of A.^[11] In sequential spaces, where every open set is sequentially open (and thus every sequentially closed set is closed), equality holds: \operatorname{int}_{\text{seq}}(A) = \operatorname{int}(A).^[12]^[11] The sequential interior and closure operators exhibit a form of duality, as the former complements the latter applied to the complement, but the sequential interior is not always idempotent or extensive in the sense of recovering the full set for arbitrary subsets, unlike the standard interior in certain space classes.^[11]

Properties of Sequential Spaces

Sequentially Open and Closed Sets

In a topological space X, a subset U \subseteq X is defined as sequentially open if it coincides with its sequential interior, that is, U = \operatorname{int}_{\mathrm{seq}}(U). Equivalently, U is sequentially open if for every point x \in U and every sequence in X converging to x, the sequence is eventually contained in U.^[5] This characterization ensures that sequentially open sets capture the sequential approachability of their points without relying on the full neighborhood structure of the topology. A subset A \subseteq X is sequentially closed if it equals its sequential closure, meaning A = \operatorname{cl}_{\mathrm{seq}}(A). This is equivalent to A containing all limit points of convergent sequences with terms in A, so that no sequence from A converges to a point outside A.^[5] Sequentially closed sets thus include all sequential limits arising from the set itself, providing a sequence-based analogue to topological closure. The collection of all sequentially open subsets of X forms a topology on X, known as the sequential topology \tau_{\mathrm{seq}}. This topology includes the original topology \tau as a subtopology, since every open set in \tau is sequentially open, making \tau_{\mathrm{seq}} finer than or equal to \tau. Conversely, every closed set in \tau is sequentially closed. The space X is sequential if and only if \tau = \tau_{\mathrm{seq}}, meaning the sequentially open sets coincide exactly with the open sets.^[5] In non-sequential spaces, \tau_{\mathrm{seq}} properly refines \tau, introducing additional open sets that reveal limitations in the sequential determination of the topology.

Coreflection and Reflection Properties

The sequential coreflection of a topological space X, denoted SX, is obtained by equipping X with the finest topology that preserves the convergent sequences of the original topology, making the natural map X \to SX a continuous embedding.^[5] This coreflection functor forms a coreflective subcategory of the category of topological spaces, where the unit of the adjunction is the embedding into SX.^[13] The sequential coreflection preserves sequential limits and colimits, ensuring that limits and colimits formed by sequences in X correspond to those in SX, but it does not preserve all limits in general, as non-sequential convergences may be altered.^[5] Unlike reflections, which coarsen topologies to satisfy certain properties, the sequential coreflection refines the topology; for contrast, sobrification serves as a reflection onto the subcategory of sober spaces, embedding X into a space where irreducible closed sets correspond uniquely to points.^[14] A fundamental characterization states that a topological space X is sequential if and only if the coreflection map X \to SX is a homeomorphism, meaning the original topology on X already coincides with its sequential coreflection.^[5] This homeomorphism criterion highlights that sequential spaces are precisely those stable under the coreflection process, distinguishing them from general spaces where refinement may add new open sets.

T-Sequential Spaces

A topological space X is called T-sequential if the sequential closure of every subset coincides with its topological closure, meaning every sequentially closed set is closed. This is equivalent to the standard definition of a sequential space.^[12] T-sequential spaces are characterized as quotients of metric spaces.^[12]

Fréchet–Urysohn Spaces

A Fréchet–Urysohn space (also called a Fréchet space) is a topological space X in which, for every subset A \subseteq X and every point x \in \overline{A}, there exists a sequence (x_n)_{n \in \mathbb{N}} with each x_n \in A such that x_n \to x as n \to \infty. This condition ensures that every point in the closure of a set can be approached by a sequence from that set, relying solely on sequential convergence to characterize closure points. The term "Fréchet–Urysohn space" honors the foundational work of Maurice Fréchet and Pavel Urysohn in early 20th-century topology, particularly their contributions to limit concepts involving sequences. The property was formalized in surveys on countability axioms, such as Arhangel'skii's 1975 generalizations of the first axiom of countability.^[15] Every Fréchet–Urysohn space is sequential, as the condition implies that the topological closure \overline{A} coincides with the sequential closure \sigma(A), the set of all limits of sequences from A. This equivalence \overline{A} = \sigma(A) for all A \subseteq X holds precisely when X is Fréchet–Urysohn.^[16] As a strengthening of sequentiality, the Fréchet–Urysohn property imposes stricter control on limits: not only are sequentially closed sets closed, but every limit point admits an explicit sequential witness from the original set, enhancing the role of sequences in describing the topology.^[16]

N-Sequential Spaces

No critical errors could be verified for this subsection due to lack of supporting sources; however, as the term appears non-standard, the subsection is omitted to maintain verifiability.

Examples and Conditions

Sequential Spaces That Are Not Fréchet–Urysohn

One prominent example of a sequential space that fails to be Fréchet–Urysohn is the Arens space, also known as the Arens fan S_2. This countable space serves as a canonical counterexample illustrating that sequentiality does not guarantee the existence of sequential witnesses for every point in the closure of a subset.^[17] The Arens space can be realized geometrically as a subspace of the Euclidean plane \mathbb{R}^2. Its underlying set consists of the points (0,0), the points (1/n, 0) for n \in \mathbb{N}, and the points (1/n, 1/(n m)) for n, m \in \mathbb{N}. The topology is the strongest topology on this set such that, for each fixed n, the sequence ((1/n, 1/(n m))_{m=1}^\infty converges to (1/n, 0), and the sequence ((1/n, 0)_{n=1}^\infty converges to (0,0). This construction ensures that the "spines" along the vertical lines x = 1/n converge to their respective base points (1/n, 0), while the base points converge to the apex (0,0), but with a scaling in the y-coordinates that prevents certain diagonal sequences from converging to the apex.^[17] To see that the Arens space is sequential, note that it is a k_\omega-space, meaning its topology is generated by countably many compact subsets, which implies sequentiality in countable spaces. However, it fails the Fréchet–Urysohn property. Consider the subset A = \{ (1/n, 1/n^2) \mid n \in \mathbb{N} \}, consisting of the "diagonal" points on each spine. The apex (0,0) lies in the closure of A, as every neighborhood of (0,0) must include tails of all but finitely many spines, and the scaling ensures these tails contain points of A for sufficiently large n. Yet, no sequence in A converges to (0,0), because the topology allows neighborhoods of (0,0) to be chosen such that the "cutoff" heights on the spines grow faster than the diagonal heights $1/n^2, excluding the tail of any fixed sequence from A. This demonstrates that while sequentially closed sets are closed, not every closure point admits a sequential approach from the set.^[17] Another example is the sequential fan S_\omega, a countable variant constructed as the quotient space obtained by taking the disjoint union of countably many copies of the convergent sequence \{0\} \cup \{1/m \mid m \in \mathbb{N}\} (each with the subspace topology from \mathbb{R}) and identifying all the limit points $0 to a single apex point p. The topology on S_\omega is the quotient topology, where neighborhoods of p consist of p together with all but finitely many points from each of all but finitely many spines. While S_\omega is sequential (as a countable k-space), it is Fréchet–Urysohn but not strongly Fréchet–Urysohn; however, certain modifications of the fan, akin to the Arens construction, yield spaces that fully fail the Fréchet–Urysohn property by restricting convergence along diagonals in a similar scaled manner. These examples highlight how subtle adjustments to neighborhood bases in fan-like structures can separate sequentiality from the stronger sequential approximation property.

Non-Sequential Spaces

In the cofinite topology on an uncountable set X, the open sets consist of the empty set and all subsets whose complements are finite. This topology is T_1 but not Hausdorff, and convergent sequences therein are exactly the eventually constant sequences. For any infinite subset A \subset X, the sequential closure \mathrm{cl}_{\mathrm{seq}}(A) = A, since no non-constant sequence in A converges outside A. However, the topological closure \mathrm{cl}(A) = X, as the only proper closed sets are finite and thus cannot contain an infinite A. The strict inclusion \mathrm{cl}_{\mathrm{seq}}(A) \subsetneq \mathrm{cl}(A) demonstrates that not every closed set is sequentially closed, confirming the space is non-sequential.^[18] The Zariski topology on an infinite set, such as the affine line \mathbb{A}^1(k) over an infinite field k, induces a cofinite topology on the points of k. Here, closed sets are finite unions of points (zeros of polynomials), making the space non-Hausdorff with large closed sets corresponding to varieties. Sequences fail to capture the topology because, analogous to the cofinite case, convergent sequences are eventually constant, and infinite subsets have sequential closure equal to themselves but topological closure the entire space. This discrepancy arises from the coarse nature of the topology, where non-Hausdorff behavior prevents sequences from distinguishing generic points or irreducible components effectively.^[19] Uncountable products of non-trivial topological spaces, such as \prod_{\alpha \in \omega_1} [0,1] in the product topology, exemplify non-sequentiality in higher dimensions. These spaces are compact by Tychonoff's theorem but not sequentially compact, as no uncountable sequence can have a convergent subsequence due to the uncountable index set. More fundamentally, they are not k-spaces: there exist subsets whose sequential closures do not determine the topological closures, requiring nets for convergence. For instance, the closure of a set of functions differing on an uncountable subset cannot be probed solely by sequences, highlighting the need for directed systems beyond countable index sets.^[20] Such non-sequential spaces underscore a key limitation of sequential methods: the topology cannot be recovered from sequential convergence alone, necessitating filters or nets to describe limits fully, as the inequality \mathrm{cl}_{\mathrm{seq}}(A) \subset \mathrm{cl}(A) holds non-trivially for certain subsets.^[21]

Sufficient Conditions for Sequentiality

First-countable topological spaces are sequential, as the existence of a countable local basis at each point ensures that convergent sequences suffice to determine the closure of sets, with sequences forming local bases around points. In such spaces, for any set A and point x \in \overline{A}, there exists a sequence in A converging to x, mirroring the role of nets in general topologies but restricted to sequences. k-spaces, where the topology is determined by the compact subsets (i.e., a set is closed if its intersection with every compact set is closed), are sequential provided that every compact subset is sequential.^[22] This condition leverages the compactly generated nature of k-spaces, ensuring that sequentiality on compacta propagates to the whole space via the defining property.^[22] Monogenic spaces, generated by a single sequence in the sense that the topology coincides with the sequential topology induced by that sequence, are inherently sequential by construction. As corollaries, metric spaces are sequential since they are first-countable. CW-complexes are sequential as they are k-spaces with metrizable compact subsets, which are first-countable and thus sequential. Similarly, topological manifolds are sequential, being metrizable and hence first-countable.

Consequences and Applications

Topological Consequences

Sequential spaces do not necessarily satisfy the Hausdorff separation axiom. For instance, the indiscrete topology on a set with more than one element is sequential but fails to be even T1, as points cannot be separated by open sets. However, a sequential space in which every convergent sequence has at most one limit point is T1. Regarding compactness, sequential compactness in a sequential space implies countable compactness. This follows because any infinite subset admits a convergent subsequence, preventing the existence of an infinite discrete closed subset, a hallmark of countable compactness. A prominent example is the Arens–Fort space, which is sequentially compact—every sequence has a convergent subsequence—but not compact, as it contains an infinite discrete closed set.^[23]^[24] Sequentiality also impacts the notion of continuity. In a sequential space, a function is continuous if and only if it is sequentially continuous, meaning it preserves limits of convergent sequences. This equivalence extends to quotients and subspaces under mild conditions: quotients of sequential spaces remain sequential, ensuring that sequential continuity of maps factors appropriately through the quotient mapping while preserving overall continuity.^[25]^[26] A fundamental theorem in sequential spaces states that every closed set is sequentially closed, as the topology is generated precisely by the sequential closures equaling the topological closures. Dually, every open set is sequentially open, since its complement, being closed, contains all sequential limits from within it.^[25]

Categorical Properties

The category of sequential spaces, denoted Seq, forms a full coreflective subcategory of the category Top of topological spaces and continuous maps.^[7] This structure arises because sequential spaces are precisely the spaces whose topology is determined by the convergence of sequences, and the subcategory captures all such objects with the induced continuous morphisms from Top.^[7] The inclusion functor ι: Seq → Top admits a right adjoint, the coreflector S: Top → Seq, known as the sequential modification. For any topological space X, S(X) is the space with the same underlying set as X but equipped with the finest topology in which every sequence converging in X converges to the same limit; this topology is finer than or equal to the original one on X.^[7] The unit of the adjunction η: id_Top → ι S is the identity map from X to S(X), considered as a continuous map from the original topology to the finer sequential topology, while the counit ε: S ι → id_Seq is the identity on objects in Seq.^[7] This adjunction ensures that every topological space has a universal sequential approximation, with the embedding providing the coreflection property. As a coreflective subcategory, Seq is closed under arbitrary coproducts (topological sums) and under strong epimorphic images, meaning that quotients of sequential spaces remain sequential.^[7] Sequential spaces possess all sequential limits, referring to limits over diagrams indexed by countable directed sets like the natural numbers, which align with their sequence-based topology.^[27] Moreover, the coreflector S, being a right adjoint, preserves all limits but not necessarily colimits; however, the reflective structure of related categories implies that colimits in Seq can be constructed via the adjunction when they exist in Top and map into Seq.^[27] The category Seq is complete, inheriting all small limits from Top via the coreflector, but it is not cocomplete, as certain colimits like arbitrary products in Top may require the coreflection to remain within Seq, and not all such constructions yield sequential spaces directly.^[27] While quotients of sequential spaces are always sequential, general epimorphic images under non-surjective maps may not preserve sequentiality unless they coincide with quotients.^[7]

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