Excluded point topology

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the excluded point topology is a <a href="/facts/Topological_space/v2ut7CbK">topology</a> where exclusion of a particular point defines <a href="/facts/Open_set/QfTCIqMu">openness</a>. Formally, let X be any non-empty set and p ∈ X. The collection

T
        =
        {
        S
        ⊆
        X
        :
        p
        ∉
        S
        }
        ∪
        {
        X
        }
      
    
    {\displaystyle T=\{S\subseteq X:p\notin S\}\cup \{X\}}

of <a href="/facts/Subset/SJGERmbA">subsets</a> of X is then the excluded point topology on X. There are a variety of cases which are individually named:

<ul><li>If X has two points, it is called the <a href="/facts/Sierpi%C5%84ski_space/Ha6nckJV">Sierpiński space</a>. This case is somewhat special and is handled separately.</li>
<li>If X is <a href="/facts/Finite_set/za1newlP">finite</a> (with at least 3 points), the topology on X is called the finite excluded point topology</li>
<li>If X is <a href="/facts/Countably_infinite/Wj4DmWPv">countably infinite</a>, the topology on X is called the countable excluded point topology</li>
<li>If X is <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a>, the topology on X is called the uncountable excluded point topology</li></ul>
A generalization is the <a href="/facts/Open_extension_topology/v090Dwsv">open extension topology</a>; if 
 
 
 
 X
 ∖
 {
 p
 }
 
 
 {\displaystyle X\setminus \{p\}}
 
 has the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a>, then the open extension topology on 
 
 
 
 (
 X
 ∖
 {
 p
 }
 )
 ∪
 {
 p
 }
 
 
 {\displaystyle (X\setminus \{p\})\cup \{p\}}
 
 is the excluded point topology. 
This topology is used to provide interesting examples and counterexamples.

Excluded point topology open-in-new

Excluded point topology