Cut-point

The "neck" of this eight-like figure is a 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. 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 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

A cut-point of a connected T₁ topological space X, is a point p in X such that X - {p} is not connected. A point which is not a cut-point is called a noncut-point.

Properties

• Cut-points are not necessarily preserved under continuous functions, (example: f: [0, 2π] → R², given by f(x) = (cos x, sin x)), but are preserved under homeomorphisms. Therefore the circle is not homeomorphic to a line segment, as the circle has no cut-points, but every point of the interval (except the two endpoints) is a cut-point.
• Every compact connected Hausdorff space, with more than one point, has at least two noncut-points.
• Every compact connected metric space, with exactly two noncut-points is homeomorphic to the unit interval.

References

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.  (Originally published by Addison-Wesley Publishing Company, Inc. in 1970.)