Arens–Fort space

The Arens- Fort space, named after the mathematicians RF Arens and MK Fort, is a specially contrived example of a topological space, which is often used due to its properties as a counterexample.

Definition

As underlying quantity we consider, ie the set of all pairs of natural numbers. The subset is called th column. We make it a topological space, the so-called Arens- Fort space by explaining the following amounts as open:

  • Any amount in that does not contain the zero point.
  • Each set containing the zero point and in almost every column almost all points ( almost here means - as usual - except for at most finitely many exceptions).

Topological properties

  • The Arens- Fort space is a normal Hausdorff space
  • Each point is a countable intersection of closed environments.
  • The Arens- Fort space is a Lindelöf space
  • Exactly the finite subsets are compact.

Missing Features

  • The Arens- Fort space satisfies neither the first nor the second axiom of countability.
  • The Arens- Fort space is not metrizable.
  • The Arens- Fort space is not compact.

Counter-examples

  • In metric spaces follows from the separability of the second axiom of countability. The Arens- Fort space shows that this is not true in general, because it is separable ( it is itself only a countable number of points ), but suffice according to the above, not the second axiom of countability.
  • Counting from the points as in Cantor's diagonal argument from the first, we obtain a sequence that always has followers in every column, and so in every neighborhood of zero.
  • Subspaces of Kelley Kelley spaces are no spaces in general. The Arens- Fort space is not a Kelley space, because the compact subsets are precisely the finite, but he is using Stone - Čech compactification of a compact subspace, and thus a Kelley space.
  • From the compact convergence does not follow the locally uniform convergence. Looking at by

Swell

  • Richard Arens: Note of Convergence in Topology. In: Mathematics Magazine. 23, 1950, ISSN 0025- 570x, pp. 229-234.
  • Lynn Arthur Steen, J. Arthur Seebach: counterexamples in topology. Second edition. Springer, New York NY, inter alia, 1978, ISBN 0-387-90312-7.
  • Topological space
  • Set topology
de