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

• Hausdorff
• regular
• normal

It is not:

• second-countable
• first-countable
• metrizable
• compact

See also

• Fort space

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