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Arens–Fort space

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Example neighborhood of (0,0) in the Arens–Fort space

In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr.

Definition

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The Arens–Fort space is the topological space where is the set of ordered pairs of non-negative integers A subset is open, that is, belongs to if and only if:

  • does not contain or
  • contains and also all but a finite number of points of all but a finite number of columns, where a column is a set with fixed.

In other words, an open set is only "allowed" to contain if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points.

Properties

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It is

It is not:

There is no sequence in that converges to However, there is a sequence in such that is a cluster point of

See also

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References

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  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446