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 p ∈ X. The collection

of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named:

A generalization / related topology is the open extension topology. That is if ${\displaystyle X\backslash \{p\}}$ 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. 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.

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