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

T = { S ⊆ X : p ∉ S } ∪ { X } {\displaystyle T=\\{S\subseteq X:p\notin S\\}\cup \\{X\\}}

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  X ∖ { p } {\displaystyle X\setminus \\{p\\}} has the discrete topology, then the open extension topology on  ( X ∖ { p } ) ∪ { p } {\displaystyle (X\setminus \\{p\\})\cup \\{p\\}} is the excluded point topology.

This topology is used to provide interesting examples and counterexamples.

## Properties

[edit]

Let  X {\displaystyle X} be a space with the excluded point topology with special point  p . {\displaystyle p.}

The space is compact, as the only neighborhood of  p {\displaystyle p} is the whole space.

The topology is an Alexandrov topology. The smallest neighborhood of  p {\displaystyle p} is the whole space  X ; {\displaystyle X;} the smallest neighborhood of a point  x ≠ p {\displaystyle x\neq p} is the singleton  { x } . {\displaystyle \\{x\\}.} These smallest neighborhoods are compact. Their closures are respectively  X {\displaystyle X} and  { x , p } , {\displaystyle \\{x,p\\},} 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  x ≠ p {\displaystyle x\neq p} do not admit a local base of closed compact neighborhoods.

The space is ultraconnected, as any nonempty closed set contains the point  p . {\displaystyle p.} Therefore the space is also connected and path-connected.

## See also

[edit]

* Finite topological space
* Fort space
* List of topologies
* Particular point topology

## References

[edit]

* 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, MR 0507446