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 edit

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 edit

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 edit

References edit

  • 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