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

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 $$x_{\bull} = \left( x_i \right)_{i=1}^{\infty}$$ in $$X \setminus \{ (0, 0) \}$$ such that $$(0, 0)$$ is a cluster point of $$x_{\bull}.$$