Let
E
E
be a Hausdorff barrelled space. If there exists a dense barrelled subspace
M
M
such that
(
codim
(
M
)
≥
c
)
[
codim
(
M
)
=
dim
(
E
)
]
(\operatorname {codim} (M) \geq c)[\operatorname {codim} (M) = \operatorname {dim} (E)]
, we say that (
M
M
is a satisfactory subspace [11]) [
E
E
is barrelledly fit], respectively. Robertson, Tweddle and Yeomans [11] proved that
E
E
has a barrelled countable enlargement (BCE) if it has a satisfactory subspace. (Trivially)
E
E
has a satisfactory subspace if
dim
(
E
)
≥
c
\dim (E) \geq c
and
E
E
is barrelledly fit. We show that
E
E
is barrelledly fit (and
dim
(
E
)
≥
c
\dim (E) \geq c
) if
E
≆
φ
E \ncong \varphi
and either (i)
E
E
is an (LF)-space, or (ii)
E
E
is an infinite-dimensional separable space and the continuum hypothesis holds. Conclusion: barrelledly fit spaces and their permanence properties arise from and advance the study of BCE’s.