Tychonoff plank

Short description: Topological space in mathematics

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 $[0,{\omega }_1]$ and $[0,\omega ]$, where $\omega$ is the first infinite ordinal and ${\omega }_1$ the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point $\mathrm{\infty }=({\omega }_1,\omega )$.

Definition

Let $\mathrm{\Omega }$ be the set of ordinals which are less than or equal to $\omega$ and ${\mathrm{\Omega }}_1$ the set of ordinals less than or equal to ${\omega }_1$. The Tychonoff plank is defined as the set $\mathrm{\Omega }\times {\mathrm{\Omega }}_1$ with the product topology.

The deleted Tychonoff plank is the subset $S=\mathrm{\Omega }\times {\mathrm{\Omega }}_1\setminus \{(\omega ,{\omega }_1)\}$, where $S$ 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 G_δ space: the singleton $\{\mathrm{\infty }\}$ is closed but not a G_δ set.

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.

See also

• List of topologies