Arens-Fort space

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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.

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

• it does not contain (0,0), or
• it contains (0,0), and all but a finite number of points in all but a finite number of columns, where a column is a set {(m,n)} with fixed m.

In other words, an open set is only "allowed" to contain (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

[edit] See also

• Fort space

[edit] References

• Counterexamples in Topology