In [1] the author showed that if there is a cardinal κ such that $2^{\kappa }={\kappa }^{+}$ then there exists a completely regular space without dense 0-dimensional subspaces. This was a solution of a problem of Arkhangel’skiĭ. 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...

Under $M{A}_{{\omega }_1}$ every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.