The construction of the Alexandroff plank starts by defining the topological space <math>(X,\tau)</math> to be the Cartesian product of <math>[0,\omega_1]</math> and <math>[-1,1],</math> where <math>\omega_1</math> is the first uncountable ordinal, and both carry the interval topology. The topology <math>\tau</math> is extended to a topology <math>\sigma</math> by adding the sets of the form
<math display=block>U(\alpha,n) = \{p\} \cup (\alpha,\omega_1] \times (0,1/n)</math>
where <math>p = (\omega_1,0) \in X.</math>
The Alexandroff plank is the topological space <math>(X,\sigma).</math>
It is called plank for being constructed from a subspace of the product of two spaces.
Properties
The space <math>(X,\sigma)</math> has the following properties:
It is Urysohn, since <math>(X,\tau)</math> is regular. The space <math>(X,\sigma)</math> is not regular, since <math>C = \{(\alpha,0) : \alpha < \omega_1\}</math> is a closed set not containing <math>(\omega_1,0),</math> while every neighbourhood of <math>C</math> intersects every neighbourhood of <math>(\omega_1,0).</math>
It is semiregular, since each basis rectangle in the topology <math>\tau</math> is a regular open set and so are the sets <math>U(\alpha,n)</math> defined above with which the topology was expanded.
It is not countably compact, since the set <math>\{(\omega_1,-1/n) : n=2,3,\dots\}</math> has no upper limit point.
It is not metacompact, since if <math>\{V_\alpha\}</math> is a covering of the ordinal space <math>[0,\omega_1)</math> with not point-finite refinement, then the covering <math>\{U_\alpha\}</math> of <math>X</math> defined by <math>U_1 = \{(0,\omega_1)\} \cup ([0,\omega_1] \times (0,1]),</math> <math>U_2 = [0,\omega_1] \times [-1,0),</math> and <math>U_\alpha = V_\alpha \times [-1,1]</math> has not point-finite refinement.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. Шаблон:ISBN (Dover edition).
S. Watson, The Construction of Topological Spaces. Recent Progress in General Topology, Elsevier, 1992.
