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 = \ \cup \$
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\setminus \$ has the discrete topology, then the open extension topology on $(X \setminus \) \cup \$ is the excluded point topology.

This topology is used to provide interesting examples and counterexamples.

Properties

Let $X$ be a space with the excluded point topology with special point $p.$

The space is compact, as the only neighborhood of $p$ is the whole space.

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

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

See also

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

References

*{{Citation , last1=Steen , first1=Lynn Arthur , author1-link=Lynn Arthur Steen , last2=Seebach , first2=J. Arthur Jr. , author2-link=J. Arthur Seebach, Jr. , title=Counterexamples in Topology , origyear=1978 , publisher=Springer-Verlag , location=Berlin, New York , edition=Dover reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995

Topological spaces