Excluded point topology
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In mathematics, the excluded point topology is a topological structure where inclusion 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 is finite (3 or more points) we call the topology on X the Finite Excluded Point topology
- If X is countable (again 3 or more points) 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
Note from the definition above the case where X has two points would be the Sierpinski space, which has very different properties then the other cases. This is the reason for requiring that X has at least 3 points.
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.
