Tychonoff Plank
المؤلف:
Kelley, J. L
المصدر:
General Topology. New York: Van Nostrand,
الجزء والصفحة:
...
7-8-2021
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Tychonoff Plank
A Tychonoff plank is a topological space that is an example of a normal space which has a non-normal subset, thus showing that normality is not a hereditary property. Let 
be the set of all ordinals which are less than or equal to 
, and 
the set of all ordinals which are less than or equal to 
. Consider the set 
with the product topology induced by the order topologies of 
and 
. Then 
is normal, but the subset ![S=Omega×Omega_1<span style=]()
{(omega,omega_1)}" src="https://mathworld.wolfram.com/images/equations/TychonoffPlank/Inline9.gif" style="height:15px; width:120px" /> is not. It can be shown that the set 
of all elements of 
whose first coordinate is equal to 
and the set 
of all elements of 
whose second coordinate is equal to 
are disjoint closed subsets 
, but there are no disjoint open subsets 
and 
of 
such that 
and 
.
REFERENCES:
Kelley, J. L. General Topology. New York: Van Nostrand, p. 132, 1955.
Willard, S. General Topology. Reading, MA: Addison-Wesley, pp. 122-123, 1970.
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