Excluded point topology explained

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 \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\setminus \{p\} has the discrete topology, then the open extension topology on (X \setminus \{p\}) \cup \{p\} 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 \{x\}. These smallest neighborhoods are compact. Their closures are respectively X and \{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\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