## 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

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## Bibliographic Information

**B. Cascales**- Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain
- Email: beca@um.es
**J. Ka̧kol**- Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Majetki 48/49,60-769 Poznań, Poland
- MR Author ID: 96980
- Email: kakol@amu.edu.pl
**S. A. Saxon**- Affiliation: Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105
- MR Author ID: 155275
- Email: saxon@math.ufl.edu
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1991777