Topological space in mathematics
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 ω -->
1
]
{\displaystyle [0,\omega _{1}]}
and
[
0
,
ω ω -->
]
{\displaystyle [0,\omega ]}
, where
ω ω -->
{\displaystyle \omega }
is the first infinite ordinal and
ω ω -->
1
{\displaystyle \omega _{1}}
the first uncountable ordinal . The deleted Tychonoff plank is obtained by deleting the point
∞ ∞ -->
=
(
ω ω -->
1
,
ω ω -->
)
{\displaystyle \infty =(\omega _{1},\omega )}
.
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.[ 2]
Notes
See also
References
Kelley, John L. (1975), General Topology , Graduate Texts in Mathematics, vol. 27 (1 ed.), New York: Springer-Verlag , Ch. 4 Ex. F, ISBN 978-0-387-90125-1 , MR 0370454
Steen, Lynn Arthur ; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag , ISBN 978-0-486-68735-3 , MR 0507446
Willard, Stephen (1970), General Topology , Addison-Wesley , 17.12 , ISBN 9780201087079 , MR 0264581