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 
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 
.
Tychonoff Plank
See also
Hereditary Property, Normal Space, Topological SpaceThis entry contributed by Margherita Barile
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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.Referenced on Wolfram|Alpha
Tychonoff PlankCite this as:
Barile, Margherita. "Tychonoff Plank." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TychonoffPlank.html