{"history":[],"article":{"id":"297519aa-1e5c-4c28-babb-e7cb7cc2ee3c","title":"Fréchet–Urysohn space","slug":"fr-chet-urysohn-space","content":":::summary\n:::\n\n# \nIn the field of [topology](https://bliptext.com/articles/topology-mathematics), a **Fréchet–Urysohn space** is a [topological space](https://bliptext.com/articles/topological-space) X with the property that for every subset the [closure](https://bliptext.com/articles/closure-topology) of S in X is identical to the sequential [closure](https://bliptext.com/articles/closure-topology) of S in X. **Fréchet–Urysohn space**s are a special type of [sequential space](https://bliptext.com/articles/sequential-space).\nThe property is named after [Maurice Fréchet](https://bliptext.com/articles/maurice-fr-chet) and [Pavel Urysohn](https://bliptext.com/articles/pavel-urysohn).\n\n# Definitions\nLet (X, \\tau) be a [topological space](https://bliptext.com/articles/topological-space).\n****T****h****e**** **** ****o****f**** ****S**** ****i****n**** ****(****X****,**** ****\\****t****a****u****)**** ****i****s**** ****t****h****e**** ****s****e****t****:****\nwhere or may be written if clarity is needed.\n****A**** ****t****o****p****o****l****o****g****i****c****a****l**** ****s****p****a****c****e**** ****(****X****,**** ****\\****t****a****u****)**** ****i****s**** ****s****a****i****d**** ****t****o**** ****b****e**** ****a**** **** ****i****f****\nfor every subset where denotes the [closure](https://bliptext.com/articles/closure-topology) of S in (X, \\tau).\n\n## Sequentially open/closed sets\n****S****u****p****p****o****s****e**** ****t****h****a****t**** ****i****s**** ****a****n****y**** ****s****u****b****s****e****t**** ****o****f**** ****X****.**** ****A**** ****s****e****q****u****e****n****c****e**** **** ****i****s**** **** ****i****f**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a**** ****p****o****s****i****t****i****v****e**** ****i****n****t****e****g****e****r**** ****N**** ****s****u****c****h**** ****t****h****a****t**** ****x****_****i**** ****\\****i****n**** ****S**** ****f****o****r**** ****a****l****l**** ****i****n****d****i****c****e****s**** ****i**** ****\\****g****e****q**** ****N****.****\n****T****h****e**** ****s****e****t**** ****S**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****e****v****e****r****y**** ****s****e****q****u****e****n****c****e**** ****i****n**** ****X**** ****t****h****a****t**** ****c****o****n****v****e****r****g****e****s**** ****t****o**** ****a**** ****p****o****i****n****t**** ****o****f**** ****S**** ****i****s**** ****e****v****e****n****t****u****a****l****l****y**** ****i****n**** ****S****;**** ****T****y****p****i****c****a****l****l****y****,**** ****i****f**** ****X**** ****i****s**** ****u****n****d****e****r****s****t****o****o****d**** ****t****h****e****n**** **** ****i****s**** ****w****r****i****t****t****e****n**** ****i****n**** ****p****l****a****c****e**** ****o****f****\n****T****h****e**** ****s****e****t**** ****S**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****o****r**** ****e****q****u****i****v****a****l****e****n****t****l****y****,**** ****i****f**** ****w****h****e****n****e****v****e****r**** **** ****i****s**** ****a**** ****s****e****q****u****e****n****c****e**** ****i****n**** ****S**** ****c****o****n****v****e****r****g****i****n****g**** ****t****o**** ****x****,**** ****t****h****e****n**** ****x**** ****m****u****s****t**** ****a****l****s****o**** ****b****e**** ****i****n**** ****S****.**** ****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****s****e****q****u****e****n****t****i****a****l****l****y**** ****o****p****e****n**** ****s****e****t**** ****i****s**** ****a**** ****s****e****q****u****e****n****t****i****a****l****l****y**** ****c****l****o****s****e****d**** ****s****e****t****,**** ****a****n****d**** ****v****i****c****e**** ****v****e****r****s****a****.****\nLet\ndenote the set of all sequentially open subsets of (X, \\tau), where this may be denoted by is the topology \\tau is understood.\nThe set is a [topology](https://bliptext.com/articles/topology-structure) on X that is [finer](https://bliptext.com/articles/comparison-of-topologies) than the original [topology](https://bliptext.com/articles/topology-structure) \\tau.\nEvery open (resp. closed) subset of X is sequentially open (resp. sequentially closed), which implies that\n\n## Strong Fréchet–Urysohn space\n****A**** ****t****o****p****o****l****o****g****i****c****a****l**** ****s****p****a****c****e**** ****X**** ****i****s**** ****a**** **** ****i****f**** ****f****o****r**** ****e****v****e****r****y**** ****p****o****i****n****t**** ****x**** ****\\****i****n**** ****X**** ****a****n****d**** ****e****v****e****r****y**** ****s****e****q****u****e****n****c****e**** ****o****f**** ****s****u****b****s****e****t****s**** ****o****f**** ****t****h****e**** ****s****p****a****c****e**** ****X**** ****s****u****c****h**** ****t****h****a****t**** **** ****t****h****e****r****e**** ****e****x****i****s****t**** ****a**** ****s****e****q****u****e****n****c****e**** **** ****i****n**** ****X**** ****s****u****c****h**** ****t****h****a****t**** ****a****_****i**** ****\\****i****n**** ****A****_****i**** ****f****o****r**** ****e****v****e****r****y**** **** ****a****n****d**** **** ****i****n**** ****(****X****,**** ****\\****t****a****u****)****.****\nThe above properties can be expressed as [selection principles](https://bliptext.com/articles/selection-principle).\n\n## Contrast to sequential spaces\nEvery open subset of X is sequentially open and every closed set is sequentially closed.\nHowever, the converses are in general not true.\n****T****h****e**** ****s****p****a****c****e****s**** ****f****o****r**** ****w****h****i****c****h**** ****t****h****e**** ****c****o****n****v****e****r****s****e****s**** ****a****r****e**** ****t****r****u****e**** ****a****r****e**** ****c****a****l****l****e****d**** ****;****\nthat is, a sequential space is a topological space in which every sequentially open subset is necessarily open, or equivalently, it is a space in which every sequentially closed subset is necessarily closed.\nEvery Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.\nSequential spaces (respectively, Fréchet-Urysohn spaces) can be viewed/interpreted as exactly those spaces X where for any single given subset knowledge of which sequences in X converge to which point(s) of X (and which do not) is sufficient to **' S is closed in X (respectively, is sufficient to **' of S in X).\nThus 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.\nIn any space that is sequential, there exists a subset for which this \"test\" gives a \"[false positive](https://bliptext.com/articles/false-positive).\"\n\n# Characterizations\nIf (X, \\tau) is a topological space then the following are equivalent:\n

X is a Fréchet–Urysohn space.
Definition: for every subset
for every subset\n
Every subspace of X is a [sequential space](https://bliptext.com/articles/sequential-space).

\nThe characterization below shows that from among Hausdorff sequential spaces, Fréchet–Urysohn spaces are exactly those for which a \"[cofinal](https://bliptext.com/articles/cofinal-set) convergent diagonal sequence\" can always be found, similar to the [diagonal principal](https://bliptext.com/articles/axiomatic-foundations-of-topological-spaces) that is used to [characterize topologies in terms of convergent nets](https://bliptext.com/articles/axiomatic-foundations-of-topological-spaces).\nIn the following characterization, all convergence is assumed to take place in (X, \\tau).\nIf (X, \\tau) is a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [sequential space](https://bliptext.com/articles/sequential-space) then X is a Fréchet–Urysohn space if and only if the following condition holds: If is a sequence in X that converge to some x \\in X and if for every l \\in \\N, is a sequence in X that converges to x_l, where these hypotheses can be summarized by the following diagram\nthen there exist strictly increasing maps such that\n(It suffices to consider only sequences with infinite ranges (i.e. is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value x, in which case the existence of the maps with the desired properties is readily verified for this special case (even if (X, \\tau) is not a Fréchet–Urysohn space).\n\n# Properties\nEvery subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.\nEvery Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.\nIf a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) (X, \\tau) is a Fréchet-Urysohn space then \\tau is equal to the [final topology](https://bliptext.com/articles/final-topology) on X induced by the set of all [arcs](https://bliptext.com/articles/arc-topology) in (X, \\tau), which by definition are continuous [paths](https://bliptext.com/articles/path-mathematics) that are also [topological embeddings](https://bliptext.com/articles/topological-embedding).\n\n# Examples\nEvery [first-countable space](https://bliptext.com/articles/first-countable-space) is a Fréchet–Urysohn space.\nConsequently, every [second-countable space](https://bliptext.com/articles/second-countable-space), every [metrizable space](https://bliptext.com/articles/metrizable-space), and every pseudo[metrizable space](https://bliptext.com/articles/metrizable-space) is a Fréchet–Urysohn space.\nIt also follows that every topological space (X, \\tau) on a finite set X is a Fréchet–Urysohn space.\n\n## Metrizable continuous dual spaces\nA [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) X (for example, a [Fréchet space](https://bliptext.com/articles/fr-chet-space)) is a [normable space](https://bliptext.com/articles/normable-space) if and only if its [strong dual space](https://bliptext.com/articles/strong-dual-space) is a Fréchet–Urysohn space, or equivalently, if and only if is a [normable space](https://bliptext.com/articles/normable-space).\n\n## Sequential spaces that are not Fréchet–Urysohn\n**Direct limit of finite-dimensional Euclidean spaces**\n**** ****\\****R****^****{****\\****i****n****f****t****y****}**** ****i****s**** ****a**** ****H****a****u****s****d****o****r****f****f**** ****s****e****q****u****e****n****t****i****a****l**** ****s****p****a****c****e**** ****t****h****a****t**** ****i****s**** ****n****o****t**** ****F****r****é****c****h****e****t****–****U****r****y****s****o****h****n****.****\nFor every integer n \\geq 1, identify \\R^n with the set where the latter is a subset of the [space of sequences](https://bliptext.com/articles/space-of-sequences) of real numbers explicitly, the elements and are identified together.\nIn particular, \\R^n can be identified as a subset of \\R^{n+1} and more generally, as a subset for any integer k \\geq 0.\nLet\nGive \\R^{\\infty} its usual topology \\tau, in which a subset is open (resp. closed) if and only if for every integer n \\geq 1, the set is an open (resp. closed) subset of \\R^n (with it usual [Euclidean topology](https://bliptext.com/articles/euclidean-topology)).\nIf and v_{\\bull} is a sequence in \\R^{\\infty} then in if and only if there exists some integer n \\geq 1 such that both v and v_{\\bull} are contained in \\R^n and in \\R^n.\nFrom these facts, it follows that is a sequential space.\nFor every integer n \\geq 1, let B_n denote the open ball in \\R^n of radius 1/n (in the [Euclidean norm](https://bliptext.com/articles/euclidean-norm)) centered at the origin.\nLet\nThen the closure of S is is all of \\R^{\\infty} but the origin of \\R^{\\infty} does belong to the sequential closure of S in In fact, it can be shown that\nThis proves that is not a Fréchet–Urysohn space.\n**Montel DF-spaces**\nEvery infinite-dimensional [Montel](https://bliptext.com/articles/montel-space) [DF-space](https://bliptext.com/articles/df-space) is a sequential space but a Fréchet–Urysohn space.\n**The [Schwartz space](https://bliptext.com/articles/schwartz-space) and the space of [smooth functions](https://bliptext.com/articles/smooth-function) **\nThe following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces.\nLet denote the [Schwartz space](https://bliptext.com/articles/schwartz-space) and let denote the space of smooth functions on an open subset where both of these spaces have their usual [Fréchet space](https://bliptext.com/articles/fr-chet-space) topologies, as defined in the article about [distributions](https://bliptext.com/articles/distribution-mathematics).\nBoth and as well as the [strong dual spaces](https://bliptext.com/articles/strong-dual-space) of both these of spaces, are [complete](https://bliptext.com/articles/complete-topological-vector-space) [nuclear](https://bliptext.com/articles/nuclear-space) [Montel](https://bliptext.com/articles/montel-space) [ultrabornological](https://bliptext.com/articles/ultrabornological-space) spaces, which implies that all four of these locally convex spaces are also [paracompact](https://bliptext.com/articles/paracompact-space) [normal](https://bliptext.com/articles/normal-space) [reflexive](https://bliptext.com/articles/reflexive-space) [barrelled spaces](https://bliptext.com/articles/barrelled-space).\nThe strong dual spaces of both and are sequential spaces but of these duals is a [Fréchet-Urysohn space](https://bliptext.com/articles/fr-chet-urysohn-space).\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T12:54:55.273Z","updated_at":"2025-01-26T12:54:55.273Z","search_vector":"'fréchet':1 'space':3 'urysohn':2","search_content":" \n \n\n# \nIn the field of [topology](https://bliptext.com/articles/topology-mathematics), a **Fréchet–Urysohn space** is a [topological space](https://bliptext.com/articles/topological-space) X with the property that for every subset the [closure](https://bliptext.com/articles/closure-topology) of S in X is identical to the sequential [closure](https://bliptext.com/articles/closure-topology) of S in X. **Fréchet–Urysohn space**s are a special type of [sequential space](https://bliptext.com/articles/sequential-space).\nThe property is named after [Maurice Fréchet](https://bliptext.com/articles/maurice-fr-chet) and [Pavel Urysohn](https://bliptext.com/articles/pavel-urysohn).\n\n# Definitions\nLet (X, \\tau) be a [topological space](https://bliptext.com/articles/topological-space).\n****T****h****e**** **** ****o****f**** ****S**** ****i****n**** ****(****X****,**** ****\\****t****a****u****)**** ****i****s**** ****t****h****e**** ****s****e****t****:****\nwhere or may be written if clarity is needed.\n****A**** ****t****o****p****o****l****o****g****i****c****a****l**** ****s****p****a****c****e**** ****(****X****,**** ****\\****t****a****u****)**** ****i****s**** ****s****a****i****d**** ****t****o**** ****b****e**** ****a**** **** ****i****f****\nfor every subset where denotes the [closure](https://bliptext.com/articles/closure-topology) of S in (X, \\tau).\n\n## Sequentially open/closed sets\n****S****u****p****p****o****s****e**** ****t****h****a****t**** ****i****s**** ****a****n****y**** ****s****u****b****s****e****t**** ****o****f**** ****X****.**** ****A**** ****s****e****q****u****e****n****c****e**** **** ****i****s**** **** ****i****f**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a**** ****p****o****s****i****t****i****v****e**** ****i****n****t****e****g****e****r**** ****N**** ****s****u****c****h**** ****t****h****a****t**** ****x****_****i**** ****\\****i****n**** ****S**** ****f****o****r**** ****a****l****l**** ****i****n****d****i****c****e****s**** ****i**** ****\\****g****e****q**** ****N****.****\n****T****h****e**** ****s****e****t**** ****S**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****e****v****e****r****y**** ****s****e****q****u****e****n****c****e**** ****i****n**** ****X**** ****t****h****a****t**** ****c****o****n****v****e****r****g****e****s**** ****t****o**** ****a**** ****p****o****i****n****t**** ****o****f**** ****S**** ****i****s**** ****e****v****e****n****t****u****a****l****l****y**** ****i****n**** ****S****;**** ****T****y****p****i****c****a****l****l****y****,**** ****i****f**** ****X**** ****i****s**** ****u****n****d****e****r****s****t****o****o****d**** ****t****h****e****n**** **** ****i****s**** ****w****r****i****t****t****e****n**** ****i****n**** ****p****l****a****c****e**** ****o****f****\n****T****h****e**** ****s****e****t**** ****S**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****o****r**** ****e****q****u****i****v****a****l****e****n****t****l****y****,**** ****i****f**** ****w****h****e****n****e****v****e****r**** **** ****i****s**** ****a**** ****s****e****q****u****e****n****c****e**** ****i****n**** ****S**** ****c****o****n****v****e****r****g****i****n****g**** ****t****o**** ****x****,**** ****t****h****e****n**** ****x**** ****m****u****s****t**** ****a****l****s****o**** ****b****e**** ****i****n**** ****S****.**** ****T****h****e**** ****c****o****m****p****l****e****m****e****n****t**** ****o****f**** ****a**** ****s****e****q****u****e****n****t****i****a****l****l****y**** ****o****p****e****n**** ****s****e****t**** ****i****s**** ****a**** ****s****e****q****u****e****n****t****i****a****l****l****y**** ****c****l****o****s****e****d**** ****s****e****t****,**** ****a****n****d**** ****v****i****c****e**** ****v****e****r****s****a****.****\nLet\ndenote the set of all sequentially open subsets of (X, \\tau), where this may be denoted by is the topology \\tau is understood.\nThe set is a [topology](https://bliptext.com/articles/topology-structure) on X that is [finer](https://bliptext.com/articles/comparison-of-topologies) than the original [topology](https://bliptext.com/articles/topology-structure) \\tau.\nEvery open (resp. closed) subset of X is sequentially open (resp. sequentially closed), which implies that\n\n## Strong Fréchet–Urysohn space\n****A**** ****t****o****p****o****l****o****g****i****c****a****l**** ****s****p****a****c****e**** ****X**** ****i****s**** ****a**** **** ****i****f**** ****f****o****r**** ****e****v****e****r****y**** ****p****o****i****n****t**** ****x**** ****\\****i****n**** ****X**** ****a****n****d**** ****e****v****e****r****y**** ****s****e****q****u****e****n****c****e**** ****o****f**** ****s****u****b****s****e****t****s**** ****o****f**** ****t****h****e**** ****s****p****a****c****e**** ****X**** ****s****u****c****h**** ****t****h****a****t**** **** ****t****h****e****r****e**** ****e****x****i****s****t**** ****a**** ****s****e****q****u****e****n****c****e**** **** ****i****n**** ****X**** ****s****u****c****h**** ****t****h****a****t**** ****a****_****i**** ****\\****i****n**** ****A****_****i**** ****f****o****r**** ****e****v****e****r****y**** **** ****a****n****d**** **** ****i****n**** ****(****X****,**** ****\\****t****a****u****)****.****\nThe above properties can be expressed as [selection principles](https://bliptext.com/articles/selection-principle).\n\n## Contrast to sequential spaces\nEvery open subset of X is sequentially open and every closed set is sequentially closed.\nHowever, the converses are in general not true.\n****T****h****e**** ****s****p****a****c****e****s**** ****f****o****r**** ****w****h****i****c****h**** ****t****h****e**** ****c****o****n****v****e****r****s****e****s**** ****a****r****e**** ****t****r****u****e**** ****a****r****e**** ****c****a****l****l****e****d**** ****;****\nthat is, a sequential space is a topological space in which every sequentially open subset is necessarily open, or equivalently, it is a space in which every sequentially closed subset is necessarily closed.\nEvery Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.\nSequential spaces (respectively, Fréchet-Urysohn spaces) can be viewed/interpreted as exactly those spaces X where for any single given subset knowledge of which sequences in X converge to which point(s) of X (and which do not) is sufficient to **' S is closed in X (respectively, is sufficient to **' of S in X).\nThus 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.\nIn any space that is sequential, there exists a subset for which this \"test\" gives a \"[false positive](https://bliptext.com/articles/false-positive).\"\n\n# Characterizations\nIf (X, \\tau) is a topological space then the following are equivalent:\n \n X is a Fréchet–Urysohn space. \n Definition: for every subset \n for every subset\n Every subspace of X is a [sequential space](https://bliptext.com/articles/sequential-space). \n**** ****F****o****r**** ****a****n****y**** ****s****u****b****s****e****t**** ****t****h****a****t**** ****i****s**** **** ****c****l****o****s****e****d**** ****i****n**** ****X**** ****a****n****d**** **** **** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a**** ****s****e****q****u****e****n****c****e**** ****i****n**** ****S**** ****t****h****a****t**** ****c****o****n****v****e****r****g****e****s**** ****t****o**** ****x****.**** **** ****\n \nThe characterization below shows that from among Hausdorff sequential spaces, Fréchet–Urysohn spaces are exactly those for which a \"[cofinal](https://bliptext.com/articles/cofinal-set) convergent diagonal sequence\" can always be found, similar to the [diagonal principal](https://bliptext.com/articles/axiomatic-foundations-of-topological-spaces) that is used to [characterize topologies in terms of convergent nets](https://bliptext.com/articles/axiomatic-foundations-of-topological-spaces).\nIn the following characterization, all convergence is assumed to take place in (X, \\tau).\nIf (X, \\tau) is a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [sequential space](https://bliptext.com/articles/sequential-space) then X is a Fréchet–Urysohn space if and only if the following condition holds: If is a sequence in X that converge to some x \\in X and if for every l \\in \\N, is a sequence in X that converges to x_l, where these hypotheses can be summarized by the following diagram\nthen there exist strictly increasing maps such that\n(It suffices to consider only sequences with infinite ranges (i.e. is infinite) because if it is finite then Hausdorffness implies that it is necessarily eventually constant with value x, in which case the existence of the maps with the desired properties is readily verified for this special case (even if (X, \\tau) is not a Fréchet–Urysohn space).\n\n# Properties\nEvery subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.\nEvery Fréchet–Urysohn space is a sequential space although the opposite implication is not true in general.\nIf a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) (X, \\tau) is a Fréchet-Urysohn space then \\tau is equal to the [final topology](https://bliptext.com/articles/final-topology) on X induced by the set of all [arcs](https://bliptext.com/articles/arc-topology) in (X, \\tau), which by definition are continuous [paths](https://bliptext.com/articles/path-mathematics) that are also [topological embeddings](https://bliptext.com/articles/topological-embedding).\n\n# Examples\nEvery [first-countable space](https://bliptext.com/articles/first-countable-space) is a Fréchet–Urysohn space.\nConsequently, every [second-countable space](https://bliptext.com/articles/second-countable-space), every [metrizable space](https://bliptext.com/articles/metrizable-space), and every pseudo[metrizable space](https://bliptext.com/articles/metrizable-space) is a Fréchet–Urysohn space.\nIt also follows that every topological space (X, \\tau) on a finite set X is a Fréchet–Urysohn space.\n\n## Metrizable continuous dual spaces\nA [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) X (for example, a [Fréchet space](https://bliptext.com/articles/fr-chet-space)) is a [normable space](https://bliptext.com/articles/normable-space) if and only if its [strong dual space](https://bliptext.com/articles/strong-dual-space) is a Fréchet–Urysohn space, or equivalently, if and only if is a [normable space](https://bliptext.com/articles/normable-space).\n\n## Sequential spaces that are not Fréchet–Urysohn\n**Direct limit of finite-dimensional Euclidean spaces**\n**** ****\\****R****^****{****\\****i****n****f****t****y****}**** ****i****s**** ****a**** ****H****a****u****s****d****o****r****f****f**** ****s****e****q****u****e****n****t****i****a****l**** ****s****p****a****c****e**** ****t****h****a****t**** ****i****s**** ****n****o****t**** ****F****r****é****c****h****e****t****–****U****r****y****s****o****h****n****.****\nFor every integer n \\geq 1, identify \\R^n with the set where the latter is a subset of the [space of sequences](https://bliptext.com/articles/space-of-sequences) of real numbers explicitly, the elements and are identified together.\nIn particular, \\R^n can be identified as a subset of \\R^{n+1} and more generally, as a subset for any integer k \\geq 0.\nLet\nGive \\R^{\\infty} its usual topology \\tau, in which a subset is open (resp. closed) if and only if for every integer n \\geq 1, the set is an open (resp. closed) subset of \\R^n (with it usual [Euclidean topology](https://bliptext.com/articles/euclidean-topology)).\nIf and v_{\\bull} is a sequence in \\R^{\\infty} then in if and only if there exists some integer n \\geq 1 such that both v and v_{\\bull} are contained in \\R^n and in \\R^n.\nFrom these facts, it follows that is a sequential space.\nFor every integer n \\geq 1, let B_n denote the open ball in \\R^n of radius 1/n (in the [Euclidean norm](https://bliptext.com/articles/euclidean-norm)) centered at the origin.\nLet\nThen the closure of S is is all of \\R^{\\infty} but the origin of \\R^{\\infty} does belong to the sequential closure of S in In fact, it can be shown that\nThis proves that is not a Fréchet–Urysohn space.\n**Montel DF-spaces**\nEvery infinite-dimensional [Montel](https://bliptext.com/articles/montel-space) [DF-space](https://bliptext.com/articles/df-space) is a sequential space but a Fréchet–Urysohn space.\n**The [Schwartz space](https://bliptext.com/articles/schwartz-space) and the space of [smooth functions](https://bliptext.com/articles/smooth-function) **\nThe following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces.\nLet denote the [Schwartz space](https://bliptext.com/articles/schwartz-space) and let denote the space of smooth functions on an open subset where both of these spaces have their usual [Fréchet space](https://bliptext.com/articles/fr-chet-space) topologies, as defined in the article about [distributions](https://bliptext.com/articles/distribution-mathematics).\nBoth and as well as the [strong dual spaces](https://bliptext.com/articles/strong-dual-space) of both these of spaces, are [complete](https://bliptext.com/articles/complete-topological-vector-space) [nuclear](https://bliptext.com/articles/nuclear-space) [Montel](https://bliptext.com/articles/montel-space) [ultrabornological](https://bliptext.com/articles/ultrabornological-space) spaces, which implies that all four of these locally convex spaces are also [paracompact](https://bliptext.com/articles/paracompact-space) [normal](https://bliptext.com/articles/normal-space) [reflexive](https://bliptext.com/articles/reflexive-space) [barrelled spaces](https://bliptext.com/articles/barrelled-space).\nThe strong dual spaces of both and are sequential spaces but of these duals is a [Fréchet-Urysohn space](https://bliptext.com/articles/fr-chet-urysohn-space).\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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