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 pX. The collection

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 has the discrete topology, then the open extension topology on 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.

See also

References

  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
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