Menu
Home Explore People Places Arts History Plants & Animals Science Life & Culture Technology
On this page
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 ⊆ X : p ∉ S } ∪ { X } {\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 X ∖ { p } {\displaystyle X\setminus \{p\}} has the discrete topology, 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.

We don't have any images related to Excluded point topology yet.
We don't have any YouTube videos related to Excluded point topology yet.
We don't have any PDF documents related to Excluded point topology yet.
We don't have any Books related to Excluded point topology yet.
We don't have any archived web articles related to Excluded point topology yet.

Properties

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

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

The topology is an Alexandrov topology. The smallest neighborhood of p {\displaystyle p} is the whole space X ; {\displaystyle X;} the smallest neighborhood of a point x ≠ p {\displaystyle x\neq p} is the singleton { x } . {\displaystyle \{x\}.} These smallest neighborhoods are compact. Their closures are respectively X {\displaystyle X} and { x , p } , {\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 x ≠ p {\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 p . {\displaystyle p.} Therefore the space is also connected and path-connected.

See also