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Concept
Excluded point topology
Summary
In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty 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:
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.
Let be a space with the excluded point topology with special point
The space is compact, as the only neighborhood of is the whole space.
The topology is an Alexandrov topology. The smallest neighborhood of is the whole space the smallest neighborhood of a point is the singleton These smallest neighborhoods are compact. Their closures are respectively and which are also 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. But points do not admit a local base of closed compact neighborhoods.
The space is ultraconnected, as any nonempty closed set contains the point Therefore the space is also connected and path-connected.
Official source
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