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If K is a Valdivia compact space, then Cp(K) is uniformly ψ-separable
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Journal Article
If K is a Valdivia compact space, then Cp(K) is uniformly ψ-separable
2023
Overview
We prove that, for any countably compact subspace
X
of a
Σ
-product of real lines, the space
C
p
(
X
)
is uniformly
ψ
-separable, that is, has a uniformly dense subset of countable pseudocharacter. This result implies that
C
p
(
K
)
is uniformly
ψ
-separable whenever
K
is a Valdivia compact space. We show that the existence of a uniformly dense realcompact subset of
C
p
(
X
)
need not imply that
C
p
(
X
)
is realcompact even if the space
X
is compact. We also establish that
C
p
(
X
)
can fail to be
ω
-monolithic if it has a uniformly dense
ω
-monolithic subspace. Furthermore, an example is given of spaces
X
and
Y
such that both
C
p
(
X
)
and
C
p
(
Y
)
are Lindelöf but
C
p
(
X
×
Y
)
has no uniformly dense Lindelöf subspace.
Publisher
Springer International Publishing
