Cut-point


In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point.
For example, every point of a line is a cut-point, while no point of a circle is a cut-point.
Cut-points are useful to determine whether two connected spaces are homeomorphic by counting the number of cut-points in each space. If two spaces have different number of cut-points, they are not homeomorphic. A classic example is using cut-points to show that lines and circles are not homeomorphic.
Cut-points are also useful in the characterization of topological continua, a class of spaces which combine the properties of compactness and connectedness and include many familiar spaces such as the unit interval, the circle, and the torus.

Definition

Formal definitions

A cut-point of a connected T1 topological space X, is a point p in X such that X - is not connected. A point which is not a cut-point is called a non-cut point.
A non-empty connected topological space X is a cut-point space if every point in X is a cut point of X.

Basic examples

Cut-points and homeomorphisms

Definitions

A cut-point space is irreducible if no proper subset of it is a cut-point space.
The Khalimsky line: Let be the set of the integers and where is a basis for a topology on. The Khalimsky line is the set endowed with this topology. It's a cut-point space. Moreover, it's irreducible.

Theorem