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.

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.

of a connected topological space

Note that these two notions only make sense if the space

A non-empty connected topological space X is called a cut-point space[2] if every point in X is a cut point of X.

A cut-point space is irreducible if no proper subset of it is a cut-point space.

Cut point (graph theory)

The "neck" of this eight-like figure is a cut-point.
A line (closed interval) has infinitely many cut points between the two endpoints. A circle has no cut point. Since they have different numbers of cut points, lines are not homeomorphic to circles.