Dense barrelled subspaces of uncountable codimension

Abstract:

Let $E$ be a Hausdorff barrelled space. If there exists a dense barrelled subspace $M$ such that $(\operatorname {codim} (M) \geq c)[\operatorname {codim} (M) = \operatorname {dim} (E)]$, we say that ($M$ is a satisfactory subspace [11]) [$E$ is barrelledly fit], respectively. Robertson, Tweddle and Yeomans [11] proved that $E$ has a barrelled countable enlargement (BCE) if it has a satisfactory subspace. (Trivially) $E$ has a satisfactory subspace if $\dim (E) \geq c$ and $E$ is barrelledly fit. We show that $E$ is barrelledly fit (and $\dim (E) \geq c$) if $E \ncong \varphi$ and either (i) $E$ is an (LF)-space, or (ii) $E$ 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.