# 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.

Let *X* be a set of ordered pairs of non-negative integers (*m*, *n*). A subset *U* of *X* is open if and only if:

- it does not contain (0, 0), or
- it contains (0, 0), and all but a finite number of points of all but a finite number of columns, where a column is a set {(
*m*,*n*)} with fixed*m*.

In other words, an open set is only "allowed" to contain (0, 0) if only a finite number of its columns contain significant gaps. By a significant gap in a column we mean the omission of an infinite number of points.

It is

It is not:

## See also

## References

- 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

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