Excluded point topology

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In mathematics, the excluded point topology is a topological structure where exclusion of a particular point defines openness. Let X be any set and . A proper subset of X is open if and only if it does not contain p. There are a variety of cases which are individually named:

• If X has two points we call it the Sierpiński space. This case is somewhat special and is handled separately.
• If X is finite (with at least 3 points) we call the topology on X the finite excluded point topology
• If X is countably infinite we call the topology on X the countable excluded point topology
• If X is uncountable we call the topology on X the uncountable excluded point topology

A generalization / related topology is the open extension topology. That is if has the discrete topology then the open extension topology will be the excluded point topology.

This topology is used to provide interesting examples and counterexamples.

[edit] See also

• Sierpiński space
• Particular point topology
• Alexandrov topology
• Finite topological space

[edit] References

• Steen, Lynn Arthur; J. Arthur Seeback (1978). Counterexamples in Topology. New York: Springer-Verlag. ISBN 0-486-68735-X (Dover edition).