Metrizability vs. Fréchet-Uryshon property

Authors:
B. Cascales, J. Kakol and S. A. Saxon

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3623-3631

MSC (2000):
Primary 54E15, 46A50

DOI:
https://doi.org/10.1090/S0002-9939-03-06944-2

Published electronically:
February 24, 2003

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Abstract: In metrizable spaces, points in the closure of a subset are limits of sequences in ; i.e., metrizable spaces are Fréchet-Uryshon spaces. The aim of this paper is to prove that metrizability and the Fréchet-Uryshon property are actually equivalent for a large class of locally convex spaces that includes - and -spaces. We introduce and study countable bounded tightness of a topological space, a property which implies countable tightness and is strictly weaker than the Fréchet-Urysohn property. We provide applications of our results to, for instance, the space of distributions . The space is not Fréchet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.

**1.**A. V. Arkhangel'skii,*Topological Function Spaces*, Kluwer Academic, Dordrecht, 1992. MR**92i:54022****2.**P. Pérez Carreras and J. Bonet,*Barrelled locally convex spaces*, North-Holland Publishing Co., Amsterdam, 1987, Notas de Matemática [Mathematical Notes], 113. MR**88j:46003****3.**B. Cascales,*On -analytic locally convex spaces*, Arch. Math. (Basel)**49**(1987), 232-244. MR**88m:46003****4.**B. Cascales, J. Kakol, and S. A. Saxon,*Weight of precompact subsets and tightness*, J. Math. Anal. Appl.**269**(2002), 500-518.**5.**B. Cascales and J. Orihuela,*On compactness in locally convex spaces*, Math. Z.**195**(1987), no. 3, 365-381. MR**88i:46021****6.**G. Choquet,*Theory of capacities*, Ann. Inst. Fourier, Grenoble**5**(1953-1954), 131-295 (1955). MR**18:295g****7.**P. Domanski and D. Vogt,*Linear topological properties of the space of analytic functions on the real line*, pp. 113-132 in*Recent Progress in Functional Analysis*, K.D. Bierstedt et al, ed., North Holland Math. Studies**189**, Amsterdam, 2001. MR**2002m:46036****8.**P. Domanski and D. Vogt,*The space of real analytic functions has no basis*, Studia Math.**142**(2001), 187-200. MR**2001m:46044****9.**S.P. Franklin,*Spaces in which sequences suffice. II.*, Fund. Math.**61**(1967), 51-56. MR**36:5882****10.**J. E. Jayne and C. A. Rogers,*Analytic sets*, ch. K-analytic sets, pp. 1-181, Academic Press, 1980.**11.**J. Kakol and S. A. Saxon,*Montel -spaces, sequential -spaces and the strongest locally convex topology*, J. London Math. Soc.,**66**(2002), 388-406.**12.**J. Kakol and S. A. Saxon,*The Fréchet-Urysohn property, -spaces and the strongest locally convex topology*, Math. Proc. Royal Irish Acad. (to appear), 2003.**13.**J. Kakol and S. A. Saxon.*The Fréchet-Urysohn view of weak and s-barrelledness*, Bull. Belgian Math. Soc. (To appear), 2002.**14.**J. Kakol and I. Tweddle,*Spaces of continuous functions as (LM)-spaces*, Bull. Belgian Math. Soc. (To appear), 2002.**15.**G. Köthe,*Topological Vector Spaces I*, Springer-Verlag, 1969. MR**40:1750****16.**P. P. Narayanaswami and S. A. Saxon,*(LF)-spaces, quasi-Baire spaces and the strongest locally convex topology*, Math. Ann.**274**(1986), no. 4, 627-641. MR**87j:46009****17.**S. A. Saxon,*Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology*, Math. Ann.**197**(1972), 87-106. MR**46:4140****18.**S. A. Saxon,*Metrizable barrelled countable enlargements*, Bull. London Math. Soc.**31**(1999), 711-718. MR**2000j:46008****19.**M. Valdivia,*Quasi-LB-spaces*, J. London Math. Soc. (2)**35**(1987), no. 1, 149-168. MR**88b:46012**

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Additional Information

**B. Cascales**

Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain

Email:
beca@um.es

**J. Kakol**

Affiliation:
Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Majetki 48/49,60-769 Poznań, Poland

Email:
kakol@amu.edu.pl

**S. A. Saxon**

Affiliation:
Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105

Email:
saxon@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-06944-2

Received by editor(s):
April 24, 2002

Received by editor(s) in revised form:
June 19, 2002

Published electronically:
February 24, 2003

Additional Notes:
The first-named author’s research was supported by D.G.E.S. grant PB 98-0381, Spain

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2003
American Mathematical Society