Alexandroff plank

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

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

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

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

The space ${\displaystyle (X,\sigma )}$ has the following properties:

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

• List of topologies – List of concrete topologies and topological spaces

• 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.