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 [math]\displaystyle{ (X,\tau) }[/math] where [math]\displaystyle{ X }[/math] is the set of ordered pairs of non-negative integers [math]\displaystyle{ (m, n). }[/math] A subset [math]\displaystyle{ U \subseteq X }[/math] is open, that is, belongs to [math]\displaystyle{ \tau, }[/math] if and only if:
- [math]\displaystyle{ U }[/math] does not contain [math]\displaystyle{ (0, 0), }[/math] or
- [math]\displaystyle{ U }[/math] contains [math]\displaystyle{ (0, 0) }[/math] and also all but a finite number of points of all but a finite number of columns, where a column is a set [math]\displaystyle{ \{ (m, n) ~:~ 0 \leq n \in \mathbb{Z} \} }[/math] with [math]\displaystyle{ 0 \leq m \in \mathbb{Z} }[/math] fixed.
In other words, an open set is only "allowed" to contain [math]\displaystyle{ (0, 0) }[/math] 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
It is not:
- second-countable
- first-countable
- metrizable
- compact
There is no sequence in [math]\displaystyle{ X \setminus \{ (0, 0) \} }[/math] that converges to [math]\displaystyle{ (0, 0). }[/math] However, there is a sequence [math]\displaystyle{ x_{\bull} = \left( x_i \right)_{i=1}^{\infty} }[/math] in [math]\displaystyle{ X \setminus \{ (0, 0) \} }[/math] such that [math]\displaystyle{ (0, 0) }[/math] is a cluster point of [math]\displaystyle{ x_{\bull}. }[/math]
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) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3
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