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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
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.

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Additional Information

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

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

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

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