In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness.
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 is the open extension topology; if
This topology is used to provide interesting examples and counterexamples.
The space is compact, as the only neighborhood of
the smallest neighborhood of a point
These smallest neighborhoods are compact.
So the space is locally relatively compact (each point admits a local base of relatively compact neighborhoods) and locally compact in the sense that each point has a local base of compact neighborhoods.
do not admit a local base of closed compact neighborhoods.
The space is ultraconnected, as any nonempty closed set contains the point