A study of the class
Δ
\Delta
consisting of topological
Δ
\Delta
-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes
Δ
2
⊂
Δ
1
\Delta _2 \subset \Delta _1
properly containing
Δ
\Delta
.
We observe that for every first-countable
X
X
the following equivalences hold:
X
∈
Δ
1
X\in \Delta _1
iff
X
∈
Δ
2
X\in \Delta _2
iff each countable subset of
X
X
is
G
δ
G_{\delta }
. Thus, new proposed concepts provide a natural extension of the family of all
λ
\lambda
-sets beyond the separable metrizable spaces.
We prove that (1) A pseudocompact space
X
X
belongs to the class
Δ
1
\Delta _1
iff countable subsets of
X
X
are scattered. (2) Every regular scattered space belongs to the class
Δ
2
\Delta _2
.
We investigate whether the classes
Δ
1
\Delta _1
and
Δ
2
\Delta _2
are invariant under the basic topological operations. Similarly to
Δ
\Delta
, both classes
Δ
1
\Delta _1
and
Δ
2
\Delta _2
are invariant under the operation of taking countable unions of closed subspaces. In contrast to
Δ
\Delta
, they are not preserved by closed continuous images.
Let
Y
Y
be
l
l
-dominated by
X
X
, i.e.
C
p
(
X
)
C_p(X)
admits a continuous linear map onto
C
p
(
Y
)
C_p(Y)
. We show that
Y
∈
Δ
1
Y \in \Delta _1
whenever
X
∈
Δ
1
X \in \Delta _1
. Moreover, we establish that if
Y
Y
is
l
l
-dominated by a compact scattered space
X
X
, then
Y
Y
is a pseudocompact space such that its Stone–Čech compactification
β
Y
\beta Y
is scattered.