Arens–Fort space

Let ${\displaystyle X}$ be a set of ordered pairs of non-negative integers ${\displaystyle (m,n).}$ A subset ${\displaystyle U\subseteq X}$ is open if and only if:

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

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

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

• Fort space
• List of topologies

• 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