  In our paper [18] we showed that a Tychonoff space $X$ is a $\Delta$-space
(in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$
is distinguished. Continuing this research, we investigate whether the class
$\Delta$ of $\Delta$-spaces is invariant under the basic topological
operations.
  We prove that if $X \in \Delta$ and $\varphi:X \to Y$ is a continuous
surjection such that $\varphi(F)$ is an $F_{\sigma}$-set in $Y$ for every
closed set $F \subset X$, then also $Y\in \Delta$. As a consequence, if $X$ is
a countable union of closed subspaces $X_i$ such that each $X_i\in \Delta$,
then also $X\in \Delta$.
  In particular, $\sigma$-product of any family of scattered Eberlein compact
spaces is a $\Delta$-space and the product of a $\Delta$-space with a countable
space is a $\Delta$-space. Our results give answers to several open problems
posed in \cite{KL}.
  Let $T:C_p(X) \longrightarrow C_p(Y)$ be a continuous linear surjection. We
observe that $T$ admits an extension to a linear continuous operator
$\widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a $\Delta$-space
whenever $X$ is.
  Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$
is a $Q$-set whenever $X$ is.
  Making use of obtained results, we provide a very short proof for the claim
that every compact $\Delta$-space has countable tightness. As a consequence,
under Proper Forcing Axiom (PFA) every compact $\Delta$-space is sequential.
  In the article we pose a dozen open questions.
