Excluded point topology
In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection
- [math]\displaystyle{ T = \{S \subseteq X : p \notin S\} \cup \{X\} }[/math]
of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named:
- If X has two points, it is called the Sierpiński space. This case is somewhat special and is handled separately.
- If X is finite (with at least 3 points), the topology on X is called the finite excluded point topology
- If X is countably infinite, the topology on X is called the countable excluded point topology
- If X is uncountable, the topology on X is called the uncountable excluded point topology
A generalization is the open extension topology; if [math]\displaystyle{ X\setminus \{p\} }[/math] has the discrete topology, then the open extension topology on [math]\displaystyle{ (X \setminus \{p\}) \cup \{p\} }[/math] is the excluded point topology.
This topology is used to provide interesting examples and counterexamples.
Properties
Let [math]\displaystyle{ X }[/math] be a space with the excluded point topology with special point [math]\displaystyle{ p. }[/math]
The space is compact, as the only neighborhood of [math]\displaystyle{ p }[/math] is the whole space.
The topology is an Alexandrov topology. The smallest neighborhood of [math]\displaystyle{ p }[/math] is the whole space [math]\displaystyle{ X; }[/math] the smallest neighborhood of a point [math]\displaystyle{ x\ne p }[/math] is the singleton [math]\displaystyle{ \{x\}. }[/math] These smallest neighborhoods are compact. Their closures are respectively [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \{x,p\}, }[/math] which are also compact. So the space is locally relatively compact (each point admits a local base of relatively compact neighborhoods) and locally compact in the sense that each point has a local base of compact neighborhoods. But points [math]\displaystyle{ x\ne p }[/math] do not admit a local base of closed compact neighborhoods.
The space is ultraconnected, as any nonempty closed set contains the point [math]\displaystyle{ p. }[/math] Therefore the space is also connected and path-connected.
See also
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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