Английская Википедия:Excluded point topology

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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

<math>T = \{S \subseteq X : p \notin S\} \cup \{X\}</math>

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 <math>X\setminus \{p\} </math> has the discrete topology, then the open extension topology on <math>(X \setminus \{p\}) \cup \{p\}</math> is the excluded point topology.

This topology is used to provide interesting examples and counterexamples.

Properties

Let <math>X</math> be a space with the excluded point topology with special point <math>p.</math>

The space is compact, as the only neighborhood of <math>p</math> is the whole space.

The topology is an Alexandrov topology. The smallest neighborhood of <math>p</math> is the whole space <math>X;</math> the smallest neighborhood of a point <math>x\ne p</math> is the singleton <math>\{x\}.</math> These smallest neighborhoods are compact. Their closures are respectively <math>X</math> and <math>\{x,p\},</math> 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 <math>x\ne p</math> do not admit a local base of closed compact neighborhoods.

The space is ultraconnected, as any nonempty closed set contains the point <math>p.</math> Therefore the space is also connected and path-connected.

See also

• Finite topological space
• Fort space
• List of topologies
• Particular point topology

References

• Шаблон:Citation