Alexandroff plank

Diagram of Alexandroff plank

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

Definition

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 $(X,\sigma )$ has the following properties:

1. It is Urysohn, since $(X,\tau )$ is regular. The space $(X,\sigma )$ is not regular, since $C=\{(\alpha ,0):\alpha <{\omega }_1\}$ is a closed set not containing $({\omega }_1,0),$ while every neighbourhood of $C$ intersects every neighbourhood of $({\omega }_1,0).$
2. It 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. It is not countably compact, since the set $\{({\omega }_1,-1/n):n=2,3,\dots \}$ has no upper limit point.
4. It 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.

See also

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References

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