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

The Arens–Fort space is the topological space $(X,\tau )$ where $X$ is the set of ordered pairs of non-negative integers $(m,n).$ A subset $U\subseteq X$ is open, that is, belongs to $\tau ,$ if and only if:

• $U$ does not contain $(0,0),$ or
• $U$ contains $(0,0)$ and also all but a finite number of points of all but a finite number of columns, where a column is a set $\{(m,n)~:~0\leq n\in \mathbb {Z} \}$ with $0\leq m\in \mathbb {Z}$ fixed.

In other words, an open set is only "allowed" to contain $(0,0)$ 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

It is

• Hausdorff
• regular
• normal
• sequential

It is not:

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

There is no sequence in $X\setminus \{(0,0)\}$ that converges to $(0,0).$ However, there is a sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X\setminus \{(0,0)\}$ such that $(0,0)$ is a cluster point of $x_{\bullet }.$

See also

• Fort space – Examples of topological spaces
• List of topologies – List of concrete topologies and topological spaces

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