Tychonoff plank

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 ${\displaystyle [0,\omega _{1}]}$ and ${\displaystyle [0,\omega ]}$, where ${\displaystyle \omega }$ is the first infinite ordinal and ${\displaystyle \omega _{1}}$ the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point ${\displaystyle \infty =(\omega _{1},\omega )}$.^[1]

Definition

Let ${\displaystyle \Omega }$ be the set of ordinals which are less than or equal to ${\displaystyle \omega }$ and ${\displaystyle \Omega _{1}}$ the set of ordinals less than or equal to ${\displaystyle \omega _{1}}$. The Tychonoff plank is defined as the set ${\displaystyle \Omega \times \Omega _{1}}$ with the product topology.^[2]

The deleted Tychonoff plank is the subset ${\displaystyle S=\Omega \times \Omega _{1}\setminus \{(\omega ,\omega _{1})\}}$, where ${\displaystyle S}$ is the plank with a corner removed.^[3]

Properties

The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal.^[4] 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 G_δ space: the singleton ${\displaystyle \{\infty \}}$ is closed but not a G_δ set.^[5]

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.^[6]

See also

• List of topologies