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 {(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.