Arens–Fort space

This article is about Arens–Fort space. Not to be confused with Arens 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.

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

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

It is not:

• second-countable
• first-countable
• metrizable
• compact
• sequential
• Fréchet–Urysohn

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