Metrizability vs. Fréchet-Uryshon property

Cascales, B.; Ka̧kol, J.; Saxon, S.

doi:10.1090/S0002-9939-03-06944-2

Metrizability vs. Fréchet-Uryshon property
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by B. Cascales, J. Ka̧kol and S. A. Saxon

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

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 $A$ are limits of sequences in $A$; 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 $(LF)$- and $(DF)$-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 $\mathfrak {D}’(\Omega )$. The space $\mathfrak {D}’(\Omega )$ is not Fréchet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.

References

A. V. Arkhangel′skiĭ, Topological function spaces, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR 1144519, DOI 10.1007/978-94-011-2598-7
Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. MR 880207
B. Cascales, On $K$-analytic locally convex spaces, Arch. Math. (Basel) 49 (1987), no. 3, 232–244. MR 906738, DOI 10.1007/BF01271663
B. Cascales, J. Ka̧kol, and S. A. Saxon, Weight of precompact subsets and tightness, J. Math. Anal. Appl. 269 (2002), 500–518.
B. Cascales and J. Orihuela, On compactness in locally convex spaces, Math. Z. 195 (1987), no. 3, 365–381. MR 895307, DOI 10.1007/BF01161762
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
P. Domański and D. Vogt, Linear topological properties of the space of analytic functions on the real line, Recent progress in functional analysis (Valencia, 2000) North-Holland Math. Stud., vol. 189, North-Holland, Amsterdam, 2001, pp. 113–132. MR 1861751, DOI 10.1016/S0304-0208(01)80040-2
PawełDomański and Dietmar Vogt, The space of real-analytic functions has no basis, Studia Math. 142 (2000), no. 2, 187–200. MR 1792604, DOI 10.4064/sm-142-2-187-200
S. P. Franklin, Spaces in which sequences suffice. II, Fund. Math. 61 (1967), 51–56. MR 222832, DOI 10.4064/fm-61-1-51-56
J. E. Jayne and C. A. Rogers, Analytic sets, ch. K-analytic sets, pp. 1–181, Academic Press, 1980.
J. Ka̧kol and S. A. Saxon, Montel $(DF)$-spaces, sequential $(LM)$-spaces and the strongest locally convex topology, J. London Math. Soc., 66 (2002), 388–406.
J. Ka̧kol and S. A. Saxon, The Fréchet-Urysohn property, $(LM)$-spaces and the strongest locally convex topology, Math. Proc. Royal Irish Acad. (to appear), 2003.
J. Ka̧kol and S. A. Saxon. The Fréchet-Urysohn view of weak and s-barrelledness, Bull. Belgian Math. Soc. (To appear), 2002.
J. Ka̧kol and I. Tweddle, Spaces of continuous functions $C_{p}(X,E)$ as (LM)-spaces, Bull. Belgian Math. Soc. (To appear), 2002.
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
P. P. Narayanaswami and Stephen A. Saxon, (LF)-spaces, quasi-Baire spaces and the strongest locally convex topology, Math. Ann. 274 (1986), no. 4, 627–641. MR 848508, DOI 10.1007/BF01458598
Stephen A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972), 87–106. MR 305010, DOI 10.1007/BF01419586
Stephen A. Saxon, Metrizable barrelled countable enlargements, Bull. London Math. Soc. 31 (1999), no. 6, 711–718. MR 1711030, DOI 10.1112/S0024609399006281
Manuel Valdivia, Quasi-LB-spaces, J. London Math. Soc. (2) 35 (1987), no. 1, 149–168. MR 871772, DOI 10.1112/jlms/s2-35.1.149