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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Metrizability vs. Fréchet-Uryshon property
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by B. Cascales, J. Ka̧kol and S. A. Saxon PDF
Proc. Amer. Math. Soc. 131 (2003), 3623-3631 Request permission

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
  • 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