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

${\displaystyle T=\{S\subseteq X:p\notin S\}\cup \{X\}}$

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

This topology is used to provide interesting examples and counterexamples.

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

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

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

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

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

• 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