In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces
[
0
,
ω
1
]
{\displaystyle [0,\omega _{1}]}
and
[
0
,
ω
]
{\displaystyle [0,\omega ]}
, where
ω
{\displaystyle \omega }
is the first infinite ordinal and
ω
1
{\displaystyle \omega _{1}}
the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point
∞
=
(
ω
1
,
ω
)
{\displaystyle \infty =(\omega _{1},\omega )}
.
Contents
Definition
Let
Ω
{\displaystyle \Omega }
be the set of ordinals which are less than or equal to
ω
{\displaystyle \omega }
and
Ω
1
{\displaystyle \Omega _{1}}
the set of ordinals less than or equal to
ω
1
{\displaystyle \omega _{1}}
. The Tychonoff plank is defined as the set
Ω
×
Ω
1
{\displaystyle \Omega \times \Omega _{1}}
with the product topology.
The deleted Tychonoff plank is the subset
S
=
Ω
×
Ω
1
∖
{
Properties
The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal. The Tychonoff plank is not perfectly normal because it is not a Gδ space: the singleton
{
∞
}
{\displaystyle \{\infty \}}
is closed but not a Gδ set.
The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.