# RACSAM. A note on Fréchet-Urysohn locally convex spaces. 1 Introduction. Jerzy Ka kol and M. López Pellicer

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2 J. Ka ol and M. López Pellicer 2 Notations, Definitions and Elementary Facts A Hausdorff topological space (space) X is said to have a compact resolution if X is covered by an ordered family {A α : α N N }, that is, such that A α A β for α β. Any K-analytic space has a compact resolution but the converse implication fails in general [19]. For a lcs X a resolution {A α : α N N } is called bounded if each set A α is bounded in X. If additionally every bounded set in X is absorbed by some K α, the resolution {A α : α N N } is called bornivorous. For α = (n ) N N put C n1n 2...n := {Kβ : β = (m l ) l, m j = n j, j = 1,..., }. Clearly K α C n1,...,n for each N. A lcs X is said to have a superresolution if for every finite tuple (n 1,... n p ) of positive integers and every bounded set Q in X there exists α = (m ) N N such that m j = n j for 1 j p and K α absorbs Q. This implies that for any finite tuple (n 1,... n p ) the sequence (C n1,...,n p,n) n is bornivorous in X. Clearly any metrizable lcs X admits a superresolution: For a countable basis (U ) of neighbourhoods of zero in X and α = (n ) N N set K α := n U. Then {K α : α N N } is as required. Moreover, (DF )-spaces, regular (LM)-spaces admit a bornivorous bounded resolution. Indeed, let X be an (LM)-space and let (X n ) n be an increasing sequence of metrizable lcs whose inductive limit is X, see [15] for details. For each n N let (U n) be a countable basis of absolutely convex neihbourhoods of zero in X n such that U n U n+1 and 2U+1 n U n for each N. For each α = (n ) N N set K α := n U n1. Then {K α : α N N } is a bounded resolution. In fact, {K α : α N N } covers X and the set n U n1 is bounded in X n1, hence in the limit space X. If additionally X is regular, i.e. for every bounded set B in X there exists m 1 N such that B is contained and bounded in X m1, then for each N there exists n N such that B n U m1. This yields a sequence α = (m ) N N such that B K α. Any lcs admitting a fundamental sequence (B n ) n of bounded sets has a superresolution; put K α := K n1 for α = (n ) N N. A space X is Fréchet-Urysohn if for each set A in X and any x A there exists a sequence of elements of A converging to x. A lcs X is barrelled (quasibarrelled) if every closed absolutely convex absorbing (bornivorous) subset of X is a neighbourhood of zero. A lcs X is called Baire-lie [17] (b-baire-lie [16]) if for every increasing (and bornivorous) sequence (A n ) n of absolutely convex closed subsets of E there exists n N such that A n is a neighbourhood of zero in E. Every metrizable (metrizable and barrelled) lcs is b-bairelie (Bairelie) and every barrelled b-baire-lie space is Baire-lie. Every Fréchet-Urysohn lcs is both b-baire-lie and bornological, [9]. By C p (X) and C c (X) we denote the spaces of real-valued continuous functions on a Tychonov space endowed with the topology of pointwise convergence and the compact-open topology, respectively. 3 Results and Remars Note that for an uncountable compact scattered space X the space C p (X) is Fréchet-Urysohn [1, Theorem III. 1. 2] non-metrizable and admits a bounded resolution. (**) Is a Fréchet-Urysohn lcs metrizable if it admits a strongly bounded resolution? Clearly every lcs with a fundamental sequence of bounded sets generates a bornivorous bounded resolution. In [3] we proved that every Fréchet-Urysohn lcs in class G is metrizable, so every Féchet-Urysohn (DF )-space is normable. Nevertheless, we have the following Example 1 Let X be a Σ-product of R I, with uncountable I, formed by all sequences of countable support. Then X contains a Fréchet-Urysohn non-metrizable vector subspace having a bornivorous bounded resolution. PROOF. Clearly X is a Fréchet-Urysohn space, see [13]. Let G be the linear span of the compact set { B := [ 1, 1] I. Set E := F G and denote B F by C. For α = (n ) set K α = n 1 C. Note that Kα : α N N} is a strongly bounded resolution in E. First observe that the family { K α : α N N} is ordered, covers E and each K α is a bounded set in E. Next, let P be a bounded set in E and let A be its closed absolutely convex cover in E. Then A is bounded in R I, hence relatively compact in R I, and 128

3 A note on Fréchet-Urysohn locally convex spaces since every cluster point of a countable set in X has countable support, A is countably compact in X. But since A is closed in E, we conclude that A is a countably compact subset of E. It is easy to see that A is a Banach dis, i.e. its linear span E A endowed with Minowsi functional norm is a Banach space. Since {nc E A : n N} is a sequence of closed absolutely convex sets in E A covering E A and E A is a Baire space, there is m N such that A mc. So if β = (n ) N N verifies that n 1 = m, then A K β, so that P K β. Therefore the family {K α : α N N } is a strongly bounded resolution in E as stated and E is non-metrizable since C is not metrizable. In general we note the following Proposition 1 For a lcs X its strong dual (X, β(x, X)) is metrizable if and only if it is Fréchet-Urysohn and X admits a bornivorous bounded resolution. PROOF. If (X, β(x, X)) is metrizable, then it is Fréchet-Urysohn and X admits a bornivorous bounded resolution. Now assume that X admits a bornivorous bounded resolution {K α : α N N }. Then the polars Kα of K α in the topological dual X of X form a base of neighbourhoods of zero for the strong topology β(x, X). But the polars Kα in X compose a resolution consisting of equicontinuous sets covering X. This shows that the space (X, β(x, X)) belongs to class G (in sense of Cascales and Orihuela of [2]). If (X, β(x, X)) is Fréchet-Urysohn, then [3, Theorem 2] yields the metrizability of (X, β(x, X)). Corollary 1 The spaces D (Ω) of distributions and A(Ω) of real analytic functions for an open set Ω R N are not Fréchret-Urysohn. PROOF. Since D (Ω) is non-metrizable (quasibarrelled) and is the strong dual of a complete (LF )-space D(Ω) of the test functions, we apply Proposition 1. The same argument can be used to the space A(Ω) via [5, Theorem 1. 7 and Proposition 1. 7]. Although (**) has a negative answer we note the following result for a large class of lcs. We need the following somewhat technical Proposition 2 A b-baire-lie space X is metrizable if and only if X admits a superresolution PROOF. Let {K α : α N N } be a superresolution for X. We may assume that all sets K α are absolutely convex. Then the sets C n1n 2...n (defined above) are also absolutely convex. First observe that for every α = (n ) N N and every neighbourhood of zero U in X there exists N such that C n1,n 2,...,n 2 U. Indeed, otherwise there exists a neighbourhood of zero U in X such that for every N there exists x C n1,n 2,...,n such that x / 2 U. Since x C n1,n 2,...,n for every N, there exists β = (m n) n N N such that x K β, n j = m j, j = 1, 2,...,. Set a n = max { m n : N } for n N and γ = (a n ) n. Since β γ for every N, then K β K γ, so x K γ for all N. Therefore 2 x 0 which provides a contradiction. The claim is proved. Let Y be the completion of X. It is clear that Y is Baire-lie. Next, we show that there exists α = (n ) N N such that C n1,n 2,...,n is a neighbourhood of zero in X for each N. Assume that does not exist n 1 N such that C n1 is a neighbourhood of zero. Since (by assumption) the sequence (nc n ) n is bornivorous in X and X is quasibarrelled, we apply [15, ] to deduce that Y = X = n nc n (1 + ɛ) n nc n. But Y is a Baire-lie space, so there exists n 1 N such that C n1 is a neighbourhood of zero in Y. Assume that for a finite tuple (n 1,... n p ) of positive integers the set C n1,...,n is a neighbourhood of zero for each 1 p. Since, by assumption, the sequence (nc n1,...,n p,n) n is bornivorous in X, and consequently X = n nc n 1,...,n p,n, we apply the same argument as above to get an integer n p+1 N such that C n1,...,n p,n p+1 is a neighbourhood of zero, which completes the inductive step. This fact combined with the claim provides a countable basis (2 C n1,...,n ) of neighbourhoods of zero, so X is metrizable. 129

4 J. Ka ol and M. López Pellicer The wea topology of any infinite-dimensional normed space is non-metrizable non-bornological but admits a superresolution. Since Fréchet-Urysohn lcs and spaces C p (X) are b-baire-lie, Proposition 2 applies to get the following Theorem 1 A Fréchet-Urysohn lcs is metrizable if and only if it admits a superresolution. C p (X) is metrizable if and only if C p (X) admits a superresolution. As we have already mentioned a Fréchet-Urysohn lcs in class G is metrizable. But only metrizable spaces C p (X) belong to class G, see [3]. On the other hand, (this fact might be already nown) under (MA)+ (CH) there exist non-metrizable analytic Fréchet-Urysohn spaces C p (X). Indeed, by [1, Theorem II.3.2] the space C p (X) is Fréchet-Urysohn if and only if X has the γ-property, i.e. if for any open cover R of X such that any finite subset of X is contained in a member of R, there exists an infinite subfamily R of R such that any element of X lies in all but finitely many members of R. Gerlits and Nagy [7] showed that under (MA) every subset of reals of cardinality smaller than the continuum has the γ-property. Hence under MA+ (CH) there are uncountable γ-subsets Y of reals, see also [8]. Thus for such Y the space C p (Y ) is non-metrizable separable and Fréchet-Urysohn. A set of reals A is said to have strong measure zero if for any sequence (t n ) n of positive reals there exists a sequence of intervals (I n ) n covering A with I n < t n for each n N. The Borel Conjecture states that every strong measure zero set is countable; Laver [11] proved that it is relatively consistent with ZFC that the Borel conjecture is true. Assuming the Borel Conjecture, every Fréchet-Urysohn separable space C p (X) is metrizable. Indeed, since C p (X) is separable and Fréchet-Urysohn, X admits a weaer separable metric topology ξ and X has the γ-property. Set Y := (X, ξ). Since Y has the γ-property, then C p (Y ) is Fréchet-Urysohn [7]. By [12, Proposition 4 and Theorem 1] the space Y is zero-dimensional. Since Y is metrizable and separable, it is homeomorphic to a subset Z with the γ-property of the Cantor set. On the other hand, Gerlits-Nagy [7] proved that γ-sets have strong measure zero. By the Borel Conjecture any γ-set in the reals is countable. Hence Z is countable. Thus X is countable and C p (X) is metrizable. Since Fréchet-Urysohn lcs are b-baire-lie, then every Fréchet-Urysohn lcs with a fundamental sequence of bounded sets must be a (DF )-space. But Fréhct-Urysohn (DF )-spaces are metrizable, [3]. For topological groups the situation is even more striing. The paper [21, Example 1.2] provides nonmetrizable Fréchet-Urysohn σ-compact topological groups but for Fréchet-Urysohn hemicompact groups X the situation is different. Next proposition extends Webb s theorem [22, Theorem 5.7], who proved that only finite-dimensional Montel (DF )-spaces enjoy the Fréchet-Urysohn property. Proposition 3 Every Fréchet-Urysohn hemicompact topological group X is a locally compact Polish space. PROOF. Let (K n ) n be an increasing sequence of compact sets covering X such that every compact set in X is contained in some K m. Observe first that X is locally compact: It is enough to show that there exists n N such that K n is a neighborhood of the unit of X. Let F be a base of neighborhoods of the unit of X and assume that no K n contains a member of F. For every U F and n N choose x U,n U \ K n, and for each n N let A n = {x U,n : U F}. Since 0 A n for every n N, there exists a sequence (U m,n ) m in F such that x m,n 0, m, where x m,n := x Um,n,n. By [14, Theorem 4] there exists a sequence (n ) of distinct numbers in N and a sequence (m ) in N such that x m,n 0. Since {x m,n : N} is contained in some K p but x m,n / K n, N, we get a contradiction. Hence X is a locally compact Fréchet-Urysohn group. Since every locally compact Fréchet-Urysohn topological group is metrizable, see e.g. [10, Theorem 2], we conclude that X is analytic. But any analytic Baire topological group is a Polish space [4, Theorem 5.4], and we reach the conclusion. Acnowledgement. This research was supported for authors by the project MTM of the Spanish Ministry of Education and Science, co-financied by the European Community (Feder funds). The first named author was also supported by the Grant MNiSW Nr. N N and Ayudas para Estancias en la U. P. V. de Investigadores de prestigio (2007). 130

5 A note on Fréchet-Urysohn locally convex spaces References [1] Arhangel sii, A. V., (1992). Topological function spaces, Math. and its Appl., Kluwer. [2] Cascales, B. and Orihuela, J., (1987). On Compactness in Locally Convex Spaces, Math. Z., 195, [3] Cascales, B., Ka ol, J. and Saxon, S. A., (2003). Metrizability vs. Fréchet-Urysohn property, Proc. Amer. Math. Soc., 131, [4] Christensen, J. P. R., (1974). Topology and Borel Structure, North Holland, Amsterdam, Math. Studies 10. [5] Domansi, P. and Vogt, D., (2001). Linear topological properties of the space of analytic functions on the real line, in Recent Progress in Functional Analysis, K. D. Bierstedt et al., eds., North-Holland Math. Studies 189, [6] Fabian, M., Habala, P., Háje, P., Montesinos, V., Pelant, J. and Zizler, V., (2001). Functional Analysis and Infinite-Dimensional Geometry, Canad. Math. Soc., Springer [7] Gerlits, J. and Nagy, Z. S., (1982). Some properties of C(X) I, Top. Appl., 14, [8] Galvin, F. and Miller, A. W., (1984). γ-sets and other singular sets of real numbers, Top. Appl., 17, [9] Ka ol, J. and Saxon, S., (2003). The Fréchet-Urysohn property, (LM)-spaces and the strongest locally convex topology, Proc. Roy. Irish Acad., 103, 1 8. [10] Ka ol, J., Lopez-Pellicer, M., Peinador, E. and Tarieladze, V., (2007). Lindelöf spaces C(X) over topological groups, Forum Math., DOI: /FORUM.2007.XXX, (to appear). [11] Laver, R.,(1976). On the consistence of Borel s conjecture, Acta Math., 137, [12] McCoy, R. A., (1980). -space function spaces, Intern. J. Math., 3, [13] Noble, N., (1970). The continuity of functions on cartesian products, Trans. Math. Soc., 149, [14] Nyios, P. J., (1981). Metrizability and Fréchet-Urysohn property in topological groups, Proc. Amer. Math. Soc., 83, [15] Pérez Carreras, P. and Bonet, J., (1987). Barrelled Locally Convex Spaces, North-Holland Math. Studies 131, Notas de Mat., 113, North-Holland, Amsterdam. [16] Ruess, W., (1976). Generalized inductive limit topologies and barrelledness properties, Pacific J. of Math., 63, [17] Saxon, A., (1972). Nuclear and product spaces, Baire-lie spaces, and the strongest locally convex topology, Math. Ann., 197, [18] Shahmatov, D., (2002). Convergence in the presence of algebraic structure, Recent Progress in Topology II, Amsterdam. [19] Talagrand, M., (1979). Espaces de Banach faiblement K-analytiques, Ann. of Math., 110, [20] Todorčević, S. and Uzcátegui, C., (2005). Analytic -spaces, Top. Appl., , [21] Todorčević, S. and Moore, J. T., (2006). The metrization problem for Fréchet groups, Preprint. [22] Webb, J. H.,(1968). Sequential convergence in locally convex spaces, Proc. Cambridge Phil. Soc., 64, Jerzy Ka ol M. López Pellicer Faculty of Mathematics and Informatics Depto. de Matemática Aplicada and IMPA A. Miciewicz University, Universidad Politécnica de Valencia Poznań, Poland E Valencia, Spain 131

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