In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space
X
X
is a
Δ
\Delta
-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex space
C
p
(
X
)
C_{p}(X)
is distinguished. Continuing this research, we investigate whether the class
Δ
\Delta
of
Δ
\Delta
-spaces is invariant under the basic topological operations.
We prove that if
X
∈
Δ
X \in \Delta
and
φ
:
X
→
Y
\varphi :X \to Y
is a continuous surjection such that
φ
(
F
)
\varphi (F)
is an
F
σ
F_{\sigma }
-set in
Y
Y
for every closed set
F
⊂
X
F \subset X
, then also
Y
∈
Δ
Y\in \Delta
. As a consequence, if
X
X
is a countable union of closed subspaces
X
i
X_i
such that each
X
i
∈
Δ
X_i\in \Delta
, then also
X
∈
Δ
X\in \Delta
. In particular,
σ
\sigma
-product of any family of scattered Eberlein compact spaces is a
Δ
\Delta
-space and the product of a
Δ
\Delta
-space with a countable space is a
Δ
\Delta
-space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99].
Let
T
:
C
p
(
X
)
⟶
C
p
(
Y
)
T:C_p(X) \longrightarrow C_p(Y)
be a continuous linear surjection. We observe that
T
T
admits an extension to a linear continuous operator
T
^
\widehat {T}
from
R
X
\mathbb {R}^X
onto
R
Y
\mathbb {R}^Y
and deduce that
Y
Y
is a
Δ
\Delta
-space whenever
X
X
is. Similarly, assuming that
X
X
and
Y
Y
are metrizable spaces, we show that
Y
Y
is a
Q
Q
-set whenever
X
X
is.
Making use of obtained results, we provide a very short proof for the claim that every compact
Δ
\Delta
-space has countable tightness. As a consequence, under Proper Forcing Axiom every compact
Δ
\Delta
-space is sequential.
In the article we pose a dozen open questions.