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 Omega be the set of all ordinals which are less than or equal to omega, and Omega_1 the set of all ordinals which are less than or equal to omega_1. Consider the set Omega×Omega_1 with the product topology induced by the order topologies of Omega and Omega_1. Then Omega×Omega_1 is normal, but the subset S=Omega×Omega_1\{(omega,omega_1)} is not. It can be shown that the set A of all elements of S whose first coordinate is equal to omega and the set B of all elements of S whose second coordinate is equal to omega_1 are disjoint closed subsets S, but there are no disjoint open subsets U and V of S such that A subset= U and B subset= V.

See also

Hereditary Property, Normal Space, Topological Space

This entry contributed by Margherita Barile

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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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Tychonoff Plank

Cite this as:

Barile, Margherita. "Tychonoff Plank." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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