Abstract
A regular topological space is called
κ-
normal if any two disjoint
κ-closed subsets in it are separated. In this paper we present some results about
κ-normality. In Section 1, the technique of adding isolated points to topological spaces has been used to construct three counterexamples. The first one shows the following statements:
(1)
A product of two linearly ordered topological spaces need not be
κ-
normal even if one of the factors is compact.
(2)
A product of two normal countably compact topological spaces need not be
κ-
normal.
(3)
A product of a normal topological space with a compact Hausdorff topological space need not be
κ-
normal.
The second one presents a mad family
R⊂[ω]
ω
such that the Mrówka space
Ψ(
R)
is not
κ-normal. The third one will show that
a scattered locally compact countably compact topological space need not be
κ-
normal. In Section 2 we have proved the following
κ-normal version of Stone's theorem:
If
X
is
κ-
normal and countably compact and
Y
is metrizable, then
X×
Y
is
κ-
normal. For a
κ-normal version of Dowker's theorem, we have been able to prove one direction which is the following statement:
If
X
is not
κ-
countably metacompact, then
X×
I
is not
κ-
normal. We use
I for the closed unit interval [0,1] with the usual topology. The converse is still unsettled. We will show that
if
X
is a Dowker space, then the Alexandroff Duplicate space
A(
X)
of
X
is a Dowker space with the property that
A(
X)×
I
is not
κ-
normal. Section 3 has been devoted to the notion of local
κ-normality. It will be shown that not every locally
κ-normal topological space is
κ-normal, even if the space satisfies other topological properties such as locally compactness, metacompactness, or countable compactness.