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 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.

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Finite topological space

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Let be a finite set.

Particular point topology

In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: If X has two points, the particular point topology on X is the Sierpiński space. If X is finite (with at least 3 points), the topology on X is called the finite particular point topology.

Topologie de Sierpiński

In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology.