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.