Arens–Fort space

Example neighborhood of (0,0) in the 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 $\{(m,n):0\le n\in \mathbb{Z}\}$ with $0\le m\in \mathbb{Z}$ 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 $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ in $X\setminus \{(0,0)\}$ such that $(0,0)$ is a cluster point of $x_{\bullet }.$

See also

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References

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