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 set and p ∈ X. The collection

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

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. A space with the excluded point topology is connected, since the only open set containing the excluded point is X itself and hence X cannot be written as disjoint union of two proper open subsets.