Linear subspace of Rl without dense totally disconnected subsets

In [1] the author showed that if there is a cardinal κ such that ${\displaystyle 2^{\kappa }={\kappa }^{+}}$ then there exists a completely regular space without dense 0-dimensional subspaces. This was a solution of a problem of Arkhangel'ski{ĭ}. Recently Arkhangel'skiĭ asked the author whether one can generalize this result by constructing a completely regular space without dense totally disconnected subspaces, and whether such a space can have a structure of a linear space. The purpose of this paper is to show that indeed such a space can be constructed under the additional assumption that there exists a cardinal κ such that ${\displaystyle 2^{\kappa }={\kappa }^{+}}$ and ${\displaystyle 2^{{\kappa }^{+}}={\kappa }^{++}}$.