Excluded point topology

In mathematics, the excluded point topology is a topological structure where exclusion of a particular point defines openness. Let "X" be any set and $pin X$ . 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 $Xackslash {p}$ 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.

ee also

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

References

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