6-cubic honeycomb

In topology, the Tychonoff plank is a topological space 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 $\mathrm{\infty }=({\omega }_1,\omega )$.

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 $\{\mathrm{\infty }\}$ is closed but not a Gδ set.

References

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