Alexandroff plank

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

Definition

Diagram of Alexandroff plank

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

The space satisfies that:

  1. 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. 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. is not countably compact, since the set has no upper limit point.
  4. 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.

References

  • 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.


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