In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces [ 0 , ω 1 ] {\displaystyle [0,\omega _{1}]} and [ 0 , ω ] {\displaystyle [0,\omega ]} , where ω {\displaystyle \omega } is the first infinite ordinal and ω 1 {\displaystyle \omega _{1}} the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point ∞ = ( ω 1 , ω ) {\displaystyle \infty =(\omega _{1},\omega )} .
Definition Let Ω {\displaystyle \Omega } be the set of ordinals which are less than or equal to ω {\displaystyle \omega } and Ω 1 {\displaystyle \Omega _{1}} the set of ordinals less than or equal to ω 1 {\displaystyle \omega _{1}} . The Tychonoff plank is defined as the set Ω × Ω 1 {\displaystyle \Omega \times \Omega _{1}} with the product topology. The deleted Tychonoff plank is the subset S = Ω × Ω 1 ∖ { ( ω , ω 1 ) } {\displaystyle S=\Omega \times \Omega _{1}\setminus \{(\omega ,\omega _{1})\}} , where S {\displaystyle S} is the plank with a corner removed.
Properties The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal. The Tychonoff plank is not perfectly normal because it is not a Gδ space: the singleton { ∞ } {\displaystyle \{\infty \}} is closed but not a Gδ set. The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.
See also List of topologies
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