Arens–Fort space explained

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 ), orU 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 Z \} with 0 \leq m \in 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) ^{inf ty} _{i =1} in X \setminus \{ ( 0 , 0 ) \} such that ( 0 , 0 ) is a cluster point of x _{\bull} .