Metrizability vs. Fréchet-Uryshon property

In metrizable spaces, points in the closure of a subset A A are limits of sequences in A 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 ( L F ) (LF) - and ( D F ) (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 D ′ ( Ω ) \mathfrak {D}’(\Omega ) . The space D ′ ( Ω ) \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.