{"history":[],"article":{"id":"8be0ca82-2e9c-440a-b658-48308a81c5cf","title":"Sequential space","slug":"sequential-space","content":":::summary\n:::\n\n# \nIn [topology](https://bliptext.com/articles/topology) and related fields of [mathematics](https://bliptext.com/articles/mathematics), a **sequential space** is a [topological space](https://bliptext.com/articles/topological-space) whose [topology](https://bliptext.com/articles/topology) can be completely characterized by its convergent/divergent sequences.\nThey can be thought of as spaces that satisfy a very weak [axiom of countability](https://bliptext.com/articles/axiom-of-countability), and all [first-countable spaces](https://bliptext.com/articles/first-countable-space) (notably [metric spaces](https://bliptext.com/articles/metric-space)) are sequential.\nIn any topological space (X, \\tau), if a convergent sequence is contained in a [closed set](https://bliptext.com/articles/closed-set) C, then the [limit](https://bliptext.com/articles/limit-mathematics) of that sequence must be contained in C as well.\nSets with this property are known as **sequentially closed**.\nSequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed.\n(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](https://bliptext.com/articles/net-mathematics) (also known as Moore–Smith sequences), but those sequences may be \"too long\" (indexed by too large an ordinal) to compress into a sequence.\nSequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.\nAny topology can be [refined](https://bliptext.com/articles/finer-topology) (that is, made finer) to a sequential topology, called the **sequential coreflection** of X.\nThe related concepts of [Fréchet–Urysohn spaces](https://bliptext.com/articles/fr-chet-urysohn-space), T-sequential spaces, and N-sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.\nSequential spaces and N-sequential spaces were introduced by [S. P. Franklin](https://bliptext.com/articles/stan-franklin).\n\n# History\nAlthough spaces satisfying such properties had implicitly been studied for several years, the first formal definition is due to S. P. Franklin in 1965.\nFranklin 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](https://bliptext.com/articles/first-countable-space), for which it was already known that sequences sufficed.\nFranklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.\n\n# Preliminary definitions\nLet X be a set and let be a [sequence](https://bliptext.com/articles/sequence) in X; that is, a family of elements of X, [indexed](https://bliptext.com/articles/indexed-family) by the [natural numbers](https://bliptext.com/articles/natural-numbers).\nIn this article, means that each element in the sequence x_{\\bull} is an element of S, and, if f : X \\to Y is a map, then For any index i, the tail of x_{\\bull} starting at i is the sequence A sequence x_{\\bull} is eventually in S if some tail of x_{\\bull} satisfies\nLet \\tau be a [topology](https://bliptext.com/articles/topological-space) on X and x_{\\bull} a sequence therein.\nThe sequence x_{\\bull} [converges](https://bliptext.com/articles/convergent-sequence) to a point x \\in X, written (when context allows, ), if, for every neighborhood U\\in\\tau of x, eventually x_{\\bull} is in U. x is then called a limit point of x_{\\bull}.\nA function f : X \\to Y between topological spaces is [sequentially continuous](https://bliptext.com/articles/sequentially-continuous) if implies\n\n# Sequential closure/interior\nLet (X, \\tau) be a topological space and let be a subset.\nThe [topological closure](https://bliptext.com/articles/closure-topology) (resp. [topological interior](https://bliptext.com/articles/interior-topology)) of S in (X, \\tau) is denoted by (resp. ).\nThe **sequential closure** of S in (X, \\tau) is the setwhich defines a map, the **sequential closure** operator, on the power set of X. If necessary for clarity, this set may also be written or It is always the case that but the reverse may fail.\nThe **sequential interior** of S in (X, \\tau) is the set(the topological space again indicated with a subscript if necessary).\nSequential closure and interior satisfy many of the nice properties of topological closure and interior: for all subsets\n

and ;\n
and ;
;
; and

\nThat is, sequential closure is a [preclosure operator](https://bliptext.com/articles/preclosure-operator).\nUnlike topological closure, sequential closure is not [idempotent](https://bliptext.com/articles/idempotence): the last containment may be strict.\nThus sequential closure is not a ([Kuratowski](https://bliptext.com/articles/kuratowski-closure-operator)) [closure operator](https://bliptext.com/articles/closure-operator).\n\n## Sequentially closed and open sets\nA set S is sequentially closed if ; equivalently, for all and x \\in X such that we must have x\\in S.\nA set S is defined to be sequentially open if its [complement](https://bliptext.com/articles/complement-set-theory) is sequentially closed.\nEquivalent conditions include:\n

or
For all and s \\in S such that eventually x_{\\bull} is in S (that is, there exists some integer i such that the tail ).

\nA set S is a **sequential neighborhood** of a point x \\in X if it contains x in its sequential interior; **sequential neighborhood**s need not be sequentially open (see below).\nIt is possible for a subset of X to be sequentially open but not open.\nSimilarly, it is possible for there to exist a sequentially closed subset that is not closed.\n\n# Sequential spaces and coreflection\nAs discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology.\nOne can obtain an idempotent sequential closure via [transfinite iteration](https://bliptext.com/articles/transfinite-iteration): for a [successor ordinal](https://bliptext.com/articles/successor-ordinal) \\alpha+1, define (as usual)and, for a [limit ordinal](https://bliptext.com/articles/limit-ordinal) \\alpha, defineThis process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index \\omega_1 (the [first uncountable ordinal](https://bliptext.com/articles/first-uncountable-ordinal)).\nConversely, the **sequential order** of X is the minimal ordinal at which, for any choice of S, the above sequence will stabilize.\nThe **transfinite sequential closure** of S is the terminal set in the above sequence: The operator is idempotent and thus a [closure operator](https://bliptext.com/articles/closure-operator).\nIn particular, it defines a topology, the sequential coreflection.\nIn the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).\n\n## Sequential spaces\nA topological space (X, \\tau) is **sequential** if it satisfies any of the following equivalent conditions:\n

\\tau is its own sequential coreflection.\n
Every sequentially open subset of X is open.
Every sequentially closed subset of X is closed.
For any subset that is closed in X, there exists some and a sequence in S that converges to x.
(Universal Property) For every topological space Y, a map f : X \\to Y is [continuous](https://bliptext.com/articles/continuous-function-topology) if and only if it is sequentially [continuous](https://bliptext.com/articles/continuous-function-topology) (if then ).\n
X is the quotient of a first-countable space.
X is the quotient of a metric space.

\nBy taking Y = X and f to be the identity map on X 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.\nIf two topologies agree on convergent sequences, then they necessarily have the same sequential coreflection.\nMoreover, a function from Y is sequentially continuous if and only if it is continuous on the sequential coreflection (that is, when pre-composed with f).\n\n# T- and N-sequential spaces\nA **T-sequential space** is a topological space with sequential order 1, which is equivalent to any of the following conditions:

The sequential closure (or interior) of every subset of X is sequentially closed (resp. open).
or are idempotent.
or
Any sequential neighborhood of x \\in X can be shrunk to a sequentially-open set that contains x; formally, sequentially-open neighborhoods are a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) for the sequential neighborhoods.
For any x \\in X and any sequential neighborhood N of x, there exists a sequential neighborhood M of x such that, for every m \\in M, the set N is a sequential neighborhood of m.

\nBeing a T-sequential space is incomparable with being a sequential space; there are sequential spaces that are not T-sequential and vice-versa.\nHowever, a topological space (X, \\tau) is called a **N-sequential** (or **neighborhood-sequential**) if it is both sequential and T-sequential.\nAn equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.\nEvery [first-countable space](https://bliptext.com/articles/first-countable-space) (and thus every [metrizable space](https://bliptext.com/articles/metrizable-space)) is N-sequential.\nThere exist [topological vector spaces](https://bliptext.com/articles/topological-vector-spaces) that are sequential but N-sequential (and thus not T-sequential).\n\n## Fréchet–Urysohn spaces\nA topological space (X, \\tau) is called [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space) if it satisfies any of the following equivalent conditions:

X is hereditarily sequential; that is, every topological subspace is sequential.\n
For every subset\n
For any subset that is not closed in X and every there exists a sequence in S that converges to x.

\nFréchet–Urysohn spaces are also sometimes said to be \"Fréchet,\" but should be confused with neither [Fréchet spaces](https://bliptext.com/articles/fr-chet-space) in [functional analysis](https://bliptext.com/articles/functional-analysis) nor the [T1 condition](https://bliptext.com/articles/t1-space).\n\n# Examples and sufficient conditions\nEvery [CW-complex](https://bliptext.com/articles/cw-complex) is sequential, as it can be considered as a quotient of a metric space.\nThe [prime spectrum](https://bliptext.com/articles/prime-spectrum) of a commutative [Noetherian ring](https://bliptext.com/articles/noetherian-ring) with the [Zariski topology](https://bliptext.com/articles/zariski-topology) is sequential.\nTake the real line \\R and [identify](https://bliptext.com/articles/quotient-space-topology) the set \\Z of integers to a point.\nAs a quotient of a metric space, the result is sequential, but it is not first countable.\nEvery [first-countable space](https://bliptext.com/articles/first-countable-space) is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential.\nThus every metrizable or [pseudometrizable](https://bliptext.com/articles/pseudometrizable-space) space — in particular, every [second-countable space](https://bliptext.com/articles/second-countable-space), [metric space](https://bliptext.com/articles/metric-space), or [discrete space](https://bliptext.com/articles/discrete-space) — is sequential.\nLet \\mathcal{F} be a set of maps from [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space) spaces to X. Then the [final topology](https://bliptext.com/articles/final-topology) that \\mathcal{F} induces on X is sequential.\nA Hausdorff [topological vector space](https://bliptext.com/articles/topological-vector-space) is sequential if and only if there exists no strictly finer topology with the same convergent sequences.\n\n## Spaces that are sequential but not Fréchet-Urysohn\n[Schwartz space](https://bliptext.com/articles/schwartz-space) and the space of [smooth functions](https://bliptext.com/articles/smooth-function), as discussed in the article on [distributions](https://bliptext.com/articles/distribution-mathematics), are both widely-used sequential spaces.\nMore generally, every infinite-dimensional [Montel](https://bliptext.com/articles/montel-space) [DF-space](https://bliptext.com/articles/df-space) is sequential but not [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space).\nArens' space is sequential, but not Fréchet–Urysohn.\n\n## Non-examples (spaces that are not sequential)\nThe simplest space that is not sequential is the [cocountable topology](https://bliptext.com/articles/cocountable-topology) on an uncountable set.\nEvery convergent sequence in such a space is eventually constant; hence every set is sequentially open.\nBut the cocountable topology is not [discrete](https://bliptext.com/articles/discrete-space).\n(One could call the topology \"sequentially discrete\".)\nLet C_c^k(U) denote the [space of](https://bliptext.com/articles/distribution-mathematics) k[-smooth test functions](https://bliptext.com/articles/distribution-mathematics) with its canonical topology and let denote the [space of](https://bliptext.com/articles/distribution-mathematics) distributions, the strong dual [space of](https://bliptext.com/articles/distribution-mathematics) ; neither are sequential (nor even an [Ascoli space](https://bliptext.com/articles/ascoli-space)).\nOn the other hand, both and are [Montel](https://bliptext.com/articles/montel-space) spaces and, in the [dual space](https://bliptext.com/articles/continuous-dual-space) of any [Montel](https://bliptext.com/articles/montel-space) space, a sequence of continuous linear functionals converges in the [strong dual topology](https://bliptext.com/articles/strong-dual-space) if and only if it converges in the [weak* topology](https://bliptext.com/articles/weak-topology) (that is, converges pointwise).\n\n# Consequences\nEvery sequential space has [countable tightness](https://bliptext.com/articles/countable-tightness) and is [compactly generated](https://bliptext.com/articles/compactly-generated-space).\nIf f : X \\to Y is a continuous [open](https://bliptext.com/articles/open-map) [surjection](https://bliptext.com/articles/surjective-map) between two Hausdorff sequential spaces then the set of points with unique preimage is closed.\n(By continuity, so is its preimage in X, the set of all points on which f is injective.)\nIf f : X \\to Y is a surjective map (not necessarily continuous) onto a Hausdorff sequential space Y and \\mathcal{B} [bases](https://bliptext.com/articles/basis-topology) for the topology on X, then f : X \\to Y is an [open map](https://bliptext.com/articles/open-map) if and only if, for every x \\in X, basic neighborhood of x, and sequence in Y, there is a subsequence of y_\\bull that is eventually in f(B).\n\n# Categorical properties\nThe [full subcategory](https://bliptext.com/articles/full-sub[category](https://bliptext.com/articles/category-mathematics)) **Seq** of all sequential spaces is closed under the following operations in the [category](https://bliptext.com/articles/category-mathematics) **Top** of topological spaces:\n• Quotients\n• Continuous closed or open images\n• Sums\n• Inductive limits\n• Open and closed subspaces\nThe category **Seq** is closed under the following operations in **Top**:\n• Continuous images\n• Subspaces\n• Finite products\nSince they are closed under topological sums and quotients, the sequential spaces form a [coreflective subcategory](https://bliptext.com/articles/coreflective-subcategory) of the [category of topological spaces](https://bliptext.com/articles/category-of-topological-spaces).\nIn fact, they are the coreflective hull of [metrizable spaces](https://bliptext.com/articles/metrizable-space) (that is, the smallest class of topological spaces closed under sums and quotients and containing the [metrizable spaces](https://bliptext.com/articles/metrizable-space)).\nThe subcategory **Seq** is a [Cartesian closed category](https://bliptext.com/articles/cartesian-closed-category) with respect to its own product (not that of **Top**).\nThe [exponential objects](https://bliptext.com/articles/exponential-object) are equipped with the (convergent sequence)-open topology.\nP.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](https://bliptext.com/articles/metric-space), [CW-complexes](https://bliptext.com/articles/cw-complex), and [differentiable manifolds](https://bliptext.com/articles/differentiable-manifold) and that is closed under colimits, quotients, and other \"certain reasonable identities\" that [Norman Steenrod](https://bliptext.com/articles/norman-steenrod) described as \"convenient\".\nEvery sequential space is [compactly generated](https://bliptext.com/articles/compactly-generated-space), and finite products in **Seq** coincide with those for [compactly generated](https://bliptext.com/articles/compactly-generated-space) spaces, since products in the category of [compactly generated](https://bliptext.com/articles/compactly-generated-space) spaces preserve quotients of metric spaces.\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:32:04.098Z","updated_at":"2025-01-26T07:32:04.098Z","search_vector":"'sequenti':1 'space':2","search_content":" \n \n\n# \nIn [topology](https://bliptext.com/articles/topology) and related fields of [mathematics](https://bliptext.com/articles/mathematics), a **sequential space** is a [topological space](https://bliptext.com/articles/topological-space) whose [topology](https://bliptext.com/articles/topology) can be completely characterized by its convergent/divergent sequences.\nThey can be thought of as spaces that satisfy a very weak [axiom of countability](https://bliptext.com/articles/axiom-of-countability), and all [first-countable spaces](https://bliptext.com/articles/first-countable-space) (notably [metric spaces](https://bliptext.com/articles/metric-space)) are sequential.\nIn any topological space (X, \\tau), if a convergent sequence is contained in a [closed set](https://bliptext.com/articles/closed-set) C, then the [limit](https://bliptext.com/articles/limit-mathematics) of that sequence must be contained in C as well.\nSets with this property are known as **sequentially closed**.\nSequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed.\n(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](https://bliptext.com/articles/net-mathematics) (also known as Moore–Smith sequences), but those sequences may be \"too long\" (indexed by too large an ordinal) to compress into a sequence.\nSequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.\nAny topology can be [refined](https://bliptext.com/articles/finer-topology) (that is, made finer) to a sequential topology, called the **sequential coreflection** of X.\nThe related concepts of [Fréchet–Urysohn spaces](https://bliptext.com/articles/fr-chet-urysohn-space), T-sequential spaces, and N-sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.\nSequential spaces and N-sequential spaces were introduced by [S. P. Franklin](https://bliptext.com/articles/stan-franklin).\n\n# History\nAlthough spaces satisfying such properties had implicitly been studied for several years, the first formal definition is due to S. P. Franklin in 1965.\nFranklin 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](https://bliptext.com/articles/first-countable-space), for which it was already known that sequences sufficed.\nFranklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.\n\n# Preliminary definitions\nLet X be a set and let be a [sequence](https://bliptext.com/articles/sequence) in X; that is, a family of elements of X, [indexed](https://bliptext.com/articles/indexed-family) by the [natural numbers](https://bliptext.com/articles/natural-numbers).\nIn this article, means that each element in the sequence x_{\\bull} is an element of S, and, if f : X \\to Y is a map, then For any index i, the tail of x_{\\bull} starting at i is the sequence A sequence x_{\\bull} is eventually in S if some tail of x_{\\bull} satisfies\nLet \\tau be a [topology](https://bliptext.com/articles/topological-space) on X and x_{\\bull} a sequence therein.\nThe sequence x_{\\bull} [converges](https://bliptext.com/articles/convergent-sequence) to a point x \\in X, written (when context allows, ), if, for every neighborhood U\\in\\tau of x, eventually x_{\\bull} is in U. x is then called a limit point of x_{\\bull}.\nA function f : X \\to Y between topological spaces is [sequentially continuous](https://bliptext.com/articles/sequentially-continuous) if implies\n\n# Sequential closure/interior\nLet (X, \\tau) be a topological space and let be a subset.\nThe [topological closure](https://bliptext.com/articles/closure-topology) (resp. [topological interior](https://bliptext.com/articles/interior-topology)) of S in (X, \\tau) is denoted by (resp. ).\nThe **sequential closure** of S in (X, \\tau) is the setwhich defines a map, the **sequential closure** operator, on the power set of X. If necessary for clarity, this set may also be written or It is always the case that but the reverse may fail.\nThe **sequential interior** of S in (X, \\tau) is the set(the topological space again indicated with a subscript if necessary).\nSequential closure and interior satisfy many of the nice properties of topological closure and interior: for all subsets\n \n and ;\n and ; \n ; \n ; and \n \n \nThat is, sequential closure is a [preclosure operator](https://bliptext.com/articles/preclosure-operator).\nUnlike topological closure, sequential closure is not [idempotent](https://bliptext.com/articles/idempotence): the last containment may be strict.\nThus sequential closure is not a ([Kuratowski](https://bliptext.com/articles/kuratowski-closure-operator)) [closure operator](https://bliptext.com/articles/closure-operator).\n\n## Sequentially closed and open sets\nA set S is sequentially closed if ; equivalently, for all and x \\in X such that we must have x\\in S.\nA set S is defined to be sequentially open if its [complement](https://bliptext.com/articles/complement-set-theory) is sequentially closed.\nEquivalent conditions include:\n \n or \n For all and s \\in S such that eventually x_{\\bull} is in S (that is, there exists some integer i such that the tail ). \n \nA set S is a **sequential neighborhood** of a point x \\in X if it contains x in its sequential interior; **sequential neighborhood**s need not be sequentially open (see below).\nIt is possible for a subset of X to be sequentially open but not open.\nSimilarly, it is possible for there to exist a sequentially closed subset that is not closed.\n\n# Sequential spaces and coreflection\nAs discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology.\nOne can obtain an idempotent sequential closure via [transfinite iteration](https://bliptext.com/articles/transfinite-iteration): for a [successor ordinal](https://bliptext.com/articles/successor-ordinal) \\alpha+1, define (as usual)and, for a [limit ordinal](https://bliptext.com/articles/limit-ordinal) \\alpha, defineThis process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index \\omega_1 (the [first uncountable ordinal](https://bliptext.com/articles/first-uncountable-ordinal)).\nConversely, the **sequential order** of X is the minimal ordinal at which, for any choice of S, the above sequence will stabilize.\nThe **transfinite sequential closure** of S is the terminal set in the above sequence: The operator is idempotent and thus a [closure operator](https://bliptext.com/articles/closure-operator).\nIn particular, it defines a topology, the sequential coreflection.\nIn the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).\n\n## Sequential spaces\nA topological space (X, \\tau) is **sequential** if it satisfies any of the following equivalent conditions:\n \n \\tau is its own sequential coreflection.\n \n Every sequentially open subset of X is open. \n Every sequentially closed subset of X is closed. \n For any subset that is closed in X, there exists some and a sequence in S that converges to x. \n (Universal Property) For every topological space Y, a map f : X \\to Y is [continuous](https://bliptext.com/articles/continuous-function-topology) if and only if it is sequentially [continuous](https://bliptext.com/articles/continuous-function-topology) (if then ).\n \n X is the quotient of a first-countable space. \n X is the quotient of a metric space. \n \nBy taking Y = X and f to be the identity map on X 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.\nIf two topologies agree on convergent sequences, then they necessarily have the same sequential coreflection.\nMoreover, a function from Y is sequentially continuous if and only if it is continuous on the sequential coreflection (that is, when pre-composed with f).\n\n# T- and N-sequential spaces\nA **T-sequential space** is a topological space with sequential order 1, which is equivalent to any of the following conditions: \n The sequential closure (or interior) of every subset of X is sequentially closed (resp. open). \n or are idempotent. \n or \n Any sequential neighborhood of x \\in X can be shrunk to a sequentially-open set that contains x; formally, sequentially-open neighborhoods are a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) for the sequential neighborhoods. \n For any x \\in X and any sequential neighborhood N of x, there exists a sequential neighborhood M of x such that, for every m \\in M, the set N is a sequential neighborhood of m. \n \nBeing a T-sequential space is incomparable with being a sequential space; there are sequential spaces that are not T-sequential and vice-versa.\nHowever, a topological space (X, \\tau) is called a **N-sequential** (or **neighborhood-sequential**) if it is both sequential and T-sequential.\nAn equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.\nEvery [first-countable space](https://bliptext.com/articles/first-countable-space) (and thus every [metrizable space](https://bliptext.com/articles/metrizable-space)) is N-sequential.\nThere exist [topological vector spaces](https://bliptext.com/articles/topological-vector-spaces) that are sequential but N-sequential (and thus not T-sequential).\n\n## Fréchet–Urysohn spaces\nA topological space (X, \\tau) is called [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space) if it satisfies any of the following equivalent conditions: \n X is hereditarily sequential; that is, every topological subspace is sequential.\n \n For every subset\n \n For any subset that is not closed in X and every there exists a sequence in S that converges to x. \n \nFréchet–Urysohn spaces are also sometimes said to be \"Fréchet,\" but should be confused with neither [Fréchet spaces](https://bliptext.com/articles/fr-chet-space) in [functional analysis](https://bliptext.com/articles/functional-analysis) nor the [T1 condition](https://bliptext.com/articles/t1-space).\n\n# Examples and sufficient conditions\nEvery [CW-complex](https://bliptext.com/articles/cw-complex) is sequential, as it can be considered as a quotient of a metric space.\nThe [prime spectrum](https://bliptext.com/articles/prime-spectrum) of a commutative [Noetherian ring](https://bliptext.com/articles/noetherian-ring) with the [Zariski topology](https://bliptext.com/articles/zariski-topology) is sequential.\nTake the real line \\R and [identify](https://bliptext.com/articles/quotient-space-topology) the set \\Z of integers to a point.\nAs a quotient of a metric space, the result is sequential, but it is not first countable.\nEvery [first-countable space](https://bliptext.com/articles/first-countable-space) is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential.\nThus every metrizable or [pseudometrizable](https://bliptext.com/articles/pseudometrizable-space) space — in particular, every [second-countable space](https://bliptext.com/articles/second-countable-space), [metric space](https://bliptext.com/articles/metric-space), or [discrete space](https://bliptext.com/articles/discrete-space) — is sequential.\nLet \\mathcal{F} be a set of maps from [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space) spaces to X. Then the [final topology](https://bliptext.com/articles/final-topology) that \\mathcal{F} induces on X is sequential.\nA Hausdorff [topological vector space](https://bliptext.com/articles/topological-vector-space) is sequential if and only if there exists no strictly finer topology with the same convergent sequences.\n\n## Spaces that are sequential but not Fréchet-Urysohn\n[Schwartz space](https://bliptext.com/articles/schwartz-space) and the space of [smooth functions](https://bliptext.com/articles/smooth-function), as discussed in the article on [distributions](https://bliptext.com/articles/distribution-mathematics), are both widely-used sequential spaces.\nMore generally, every infinite-dimensional [Montel](https://bliptext.com/articles/montel-space) [DF-space](https://bliptext.com/articles/df-space) is sequential but not [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space).\nArens' space is sequential, but not Fréchet–Urysohn.\n\n## Non-examples (spaces that are not sequential)\nThe simplest space that is not sequential is the [cocountable topology](https://bliptext.com/articles/cocountable-topology) on an uncountable set.\nEvery convergent sequence in such a space is eventually constant; hence every set is sequentially open.\nBut the cocountable topology is not [discrete](https://bliptext.com/articles/discrete-space).\n(One could call the topology \"sequentially discrete\".)\nLet C_c^k(U) denote the [space of](https://bliptext.com/articles/distribution-mathematics) k[-smooth test functions](https://bliptext.com/articles/distribution-mathematics) with its canonical topology and let denote the [space of](https://bliptext.com/articles/distribution-mathematics) distributions, the strong dual [space of](https://bliptext.com/articles/distribution-mathematics) ; neither are sequential (nor even an [Ascoli space](https://bliptext.com/articles/ascoli-space)).\nOn the other hand, both and are [Montel](https://bliptext.com/articles/montel-space) spaces and, in the [dual space](https://bliptext.com/articles/continuous-dual-space) of any [Montel](https://bliptext.com/articles/montel-space) space, a sequence of continuous linear functionals converges in the [strong dual topology](https://bliptext.com/articles/strong-dual-space) if and only if it converges in the [weak* topology](https://bliptext.com/articles/weak-topology) (that is, converges pointwise).\n\n# Consequences\nEvery sequential space has [countable tightness](https://bliptext.com/articles/countable-tightness) and is [compactly generated](https://bliptext.com/articles/compactly-generated-space).\nIf f : X \\to Y is a continuous [open](https://bliptext.com/articles/open-map) [surjection](https://bliptext.com/articles/surjective-map) between two Hausdorff sequential spaces then the set of points with unique preimage is closed.\n(By continuity, so is its preimage in X, the set of all points on which f is injective.)\nIf f : X \\to Y is a surjective map (not necessarily continuous) onto a Hausdorff sequential space Y and \\mathcal{B} [bases](https://bliptext.com/articles/basis-topology) for the topology on X, then f : X \\to Y is an [open map](https://bliptext.com/articles/open-map) if and only if, for every x \\in X, basic neighborhood of x, and sequence in Y, there is a subsequence of y_\\bull that is eventually in f(B).\n\n# Categorical properties\nThe [full sub[category](https://bliptext.com/articles/category-mathematics)](https://bliptext.com/articles/full-sub[category](https://bliptext.com/articles/category-mathematics)) **Seq** of all sequential spaces is closed under the following operations in the [category](https://bliptext.com/articles/category-mathematics) **Top** of topological spaces:\n• Quotients\n• Continuous closed or open images\n• Sums\n• Inductive limits\n• Open and closed subspaces\nThe category **Seq** is closed under the following operations in **Top**:\n• Continuous images\n• Subspaces\n• Finite products\nSince they are closed under topological sums and quotients, the sequential spaces form a [coreflective subcategory](https://bliptext.com/articles/coreflective-subcategory) of the [category of topological spaces](https://bliptext.com/articles/category-of-topological-spaces).\nIn fact, they are the coreflective hull of [metrizable spaces](https://bliptext.com/articles/metrizable-space) (that is, the smallest class of topological spaces closed under sums and quotients and containing the [metrizable spaces](https://bliptext.com/articles/metrizable-space)).\nThe subcategory **Seq** is a [Cartesian closed category](https://bliptext.com/articles/cartesian-closed-category) with respect to its own product (not that of **Top**).\nThe [exponential objects](https://bliptext.com/articles/exponential-object) are equipped with the (convergent sequence)-open topology.\nP.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](https://bliptext.com/articles/metric-space), [CW-complexes](https://bliptext.com/articles/cw-complex), and [differentiable manifolds](https://bliptext.com/articles/differentiable-manifold) and that is closed under colimits, quotients, and other \"certain reasonable identities\" that [Norman Steenrod](https://bliptext.com/articles/norman-steenrod) described as \"convenient\".\nEvery sequential space is [compactly generated](https://bliptext.com/articles/compactly-generated-space), and finite products in **Seq** coincide with those for [compactly generated](https://bliptext.com/articles/compactly-generated-space) spaces, since products in the category of [compactly generated](https://bliptext.com/articles/compactly-generated-space) spaces preserve quotients of metric spaces.\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}