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 Space
*This 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 Plank## Cite this as:

Barile, Margherita. "Tychonoff Plank." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/TychonoffPlank.html