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

Diagram of Alexandroff plank

Definition

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The construction of the Alexandroff plank starts by defining the topological space   to be the Cartesian product of   and   where   is the first uncountable ordinal, and both carry the interval topology. The topology   is extended to a topology   by adding the sets of the form   where  

The Alexandroff plank is the topological space  

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

Properties

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The space   has the following properties:

  1. It is Urysohn, since   is regular. The space   is not regular, since   is a closed set not containing   while every neighbourhood of   intersects every neighbourhood of  
  2. It is semiregular, since each basis rectangle in the topology   is a regular open set and so are the sets   defined above with which the topology was expanded.
  3. It is not countably compact, since the set   has no upper limit point.
  4. It is not metacompact, since if   is a covering of the ordinal space   with not point-finite refinement, then the covering   of   defined by     and   has not point-finite refinement.

See also

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References

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  • 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.