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

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.