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.