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 Ω be the set of all ordinals which are less than or equal to ω, and Ω_1 the set of all ordinals which are less than or equal to ω_1. Consider the set Ω×Ω_1 with the product topology induced by the order topologies of Ω and Ω_1. Then Ω×Ω_1 is normal, but the subset S = Ω×Ω_1 \{(ω, ω_1)} is not. It can be shown that the set A of all elements of S whose first coordinate is equal to ω and the set B of all elements of S whose second coordinate is equal to ω_1 are disjoint closed subsets S, but there are no disjoint open subsets U and V of S such that A⊆U and B⊆V.

hereditary property | normal space | topological space