  $\Delta$-spaces have been defined by a natural generalization of a classical
notion of $\Delta$-sets of reals to Tychonoff topological spaces; moreover, the
class $\Delta$ of all $\Delta$-spaces consists precisely of those $X$ for which
the locally convex space $C_p(X)$ is distinguished.
  The aim of this article is to better understand the boundaries of the class
$\Delta$, by presenting new examples and counter-examples.
  1) We examine when trees considered as topological spaces equipped with the
interval topology belong to $\Delta$. In particular, we prove that no Souslin
tree is a $\Delta$-space. Other main results are connected with the study of 2)
$\Psi$-spaces built on maximal almost disjoint families of countable sets; and
3) Ladder system spaces.
  It is consistent with CH that all ladder system spaces on $\omega_1$ are in
$\Delta$. We show that in forcing extension of ZFC obtained by adding one Cohen
real, there is a ladder system space on $\omega_1$ which is not in $\Delta$.
  We resolve several open problems posed in the literature.
