Arens–Fort space

Short description: Topological space

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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.

## Contents

* 1 Definition
* 2 Properties
* 3 See also
* 4 References

## Definition

The Arens–Fort space is the topological space (X,τ) where X is the set of ordered pairs of non-negative integers (m,n). A subset U⊆X is open, that is, belongs to τ, 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≤n∈ℤ} with 0≤m∈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∖{(0,0)} that converges to (0,0). However, there is a sequence x∙=(xi)i=1∞ in X∖{(0,0)} such that (0,0) is a cluster point of x∙.

## See also

* Fort space - Examples of topological spaces
* List of topologies - List of concrete topologies and topological spaces

## References

* Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995), Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3

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