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 point p of a connected topological space X is called a cut point of X if is not connected. A point p of a connected space X is called a non-cut point of X if is connected. Note that these two notions only make sense if the space X is connected to start with. Also, for a space to have a cut point, the space must have at least three points, because removing a point from a space with one or two elements always leaves a connected space. A non-empty connected topological space X is called a cut-point space if every point in X is a cut point of X.

Basic examples

Notations

Theorems

Cut-points and homeomorphisms

Cut-points and continua

Topological properties of cut-point spaces

Irreducible cut-point spaces

Definitions

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

Theorem