Alexandroff plank

Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.

Definition

Diagram of Alexandroff plank

The construction of the Alexandroff plank starts by defining the topological space $(X,\tau )$ to be the Cartesian product of $[0,\omega _{1}]$ and $[-1,1]$, where $\omega _{1}$ is the first uncountable ordinal, and both carry the interval topology. The topology $\tau$ is extended to a topology $\sigma$ by adding the sets of the form

$U(\alpha ,n)=\{p\}\cup (\alpha ,\omega _{1}]\times (0,1/n)$

where $p=(\omega _{1},0)\in X$.

The Alexandroff plank is the topological space $(X,\sigma )$.

It is called plank for being constructed from a subspace of the product of two spaces.

Properties

The space $(X,\sigma )$ satisfies that:

1. is Urysohn, since $(X,\tau )$ is regular. The space $(X,\sigma )$ is not regular, since $C=\{(\alpha ,0)\mid \alpha <\omega _{1}\}$ is a closed set not containing $(\omega _{1},0)$, while every neighbourhood of $C$ intersects every neighbourhood of $(\omega _{1},0)$.
2. is semiregular, since each basis rectangle in the topology $\tau$ is a regular open set and so are the sets $U(\alpha ,n)$ defined above with which the topology was expanded.
3. is not countably compact, since the set $\{(\omega _{1},-1/n)\mid n=2,3,\ldots \}$ has no upper limit point.
4. is not metacompact, since if $\{V_{\alpha }\}$ is a covering of the ordinal space $[0,\omega _{1})$ with not point-finite refinement, then the covering $\{U_{\alpha }\}$ of $X$ defined by $U_{1}=\{(0,\omega _{1})\}\cup ([0,\omega _{1}]\times (0,1])$, $U_{2}=[0,\omega _{1}]\times [-1,0)$, and $U_{\alpha }=V_{\alpha }\times [-1,1]$ has not point-finite refinement.

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
• S. Watson, The Construction of Topological Spaces. Recent Progress in General Topology, Elsevier, 1992.