A space
X
X
is
>
α
> \alpha
-bounded if for all
A
⊆
X
A \subseteq X
with
|
A
|
>
α
|A| > \alpha
,
cl
X
A
{\operatorname {cl} _X}\;A
is compact. Let
B
(
α
)
B(\alpha )
be the smallest
>
α
> \alpha
-bounded subspace of
β
(
α
)
\beta (\alpha )
containing
α
\alpha
. It is shown that the following properties are equivalent: (a)
α
\alpha
is a singular cardinal; (b)
B
(
α
)
B(\alpha )
is not locally compact; (c)
B
(
α
)
B(\alpha )
is
α
\alpha
-pseudocompact; (d)
B
(
α
)
B(\alpha )
is initially
α
\alpha
-compact. Define
B
0
(
α
)
=
α
{B^0}(\alpha ) = \alpha
and
B
n
(
α
)
=
{
cl
β
(
α
)
A
:
A
⊆
B
n
−
1
(
α
)
,
|
A
|
>
α
}
{B^n}(\alpha ) = \{ {\operatorname {cl} _{\beta (\alpha )}}A:A \subseteq {B^{n - 1}}(\alpha ),|A| > \alpha \}
for
0
>
n
>
ω
0 > n > \omega
. We also prove that
B
2
(
α
)
≠
B
3
(
α
)
{B^2}(\alpha ) \ne {B^3}(\alpha )
when
ω
=
cf
(
α
)
>
α
\omega = \operatorname {cf} (\alpha ) > \alpha
. Finally, we calculate the cardinality of
B
(
α
)
B(\alpha )
and prove that, for every singular cardinal
α
,
|
B
(
α
)
|
=
|
B
(
α
)
|
α
=
|
N
(
α
)
|
cf
(
α
)
\alpha ,\;|B(\alpha )| = |B(\alpha ){|^\alpha } = |N(\alpha ){|^{\operatorname {cf} (\alpha )}}
where
N
(
α
)
=
{
p
∈
β
(
α
)
:
there is
A
∈
p
with
|
A
|
>
α
}
N(\alpha ) = \{ p \in \beta (\alpha ):\;{\text {there is}}\;A \in p\;{\text {with}}\;|A| > \alpha \}
.