In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the excluded point topology is a topology

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 where exclusion of a particular point defines openness

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

. Formally, let X be any set and p ∈ X. The collection

T = {S ⊆ X: p ∉ S or S = X;}

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 we call it the Sierpiński space
  Sierpinski space
  In mathematics, the Sierpiński space 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...
  
  . This case is somewhat special and is handled separately.
• If X is finite (with at least 3 points) we call the topology on X the finite excluded point topology
• If X is countably infinite 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

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.

See also

• Sierpiński space
  Sierpinski space
  In mathematics, the Sierpiński space 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...
  
  
• Particular point topology
  Particular point topology
  In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
  
  
• Alexandrov topology
  Alexandrov topology
  In topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open...
  
  
• Finite topological space
  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 for which there are only finitely many points....
  
  
• Fort space
  Fort space
  In mathematics, Fort space, named after M. K. Fort, Jr., is an example in the theory of topological spaces.Let X be an infinite set of points, of which P is one...