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 T1topological spaceX, 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 topologicalspace X is a cut-point space if every point in X is a cut point of X.
Basic examples
A closed interval has infinitely many cut-points. All points except for its end points are cut-points and the end-points are non-cut points.
An open interval also has infinitely many cut-points like closed intervals. Since open intervals don't have end-points, it has no non-cut points.
A circle has no cut-points and it follows that every point of a circle is a non-cut point.
Notations
A cutting of X is a set where p is a cut-point of X, U and V form a separation of X-.
Also can be written as X\=U|V.
Theorems
Cut-points and homeomorphisms
Cut-points are not necessarily preserved under continuous functions. For example: f: → R2, given by f =. Every point of the interval is a cut-point, but f forms a circle which has no cut-points.
Cut-points are preserved under homeomorphisms. Therefore, cut-point is a topological invariant.
Cut-points and continua
Every continuum with more than one point, has at least two non-cut points. Specifically, each open set which forms a separation of resulting space contains at least one non-cut point.
Every continuum with exactly two noncut-points is homeomorphic to the unit interval.
If K is a continuum with points a,b and K- isn't connected, K is homeomorphic to the unit circle.
Let X be a connected space and x be a cut point in X such that X\=A|B. Then is either open or closed. if is open, A and B are closed. If is closed, A and B are open.
Let X be a cut-point space. The set of closed points of X is infinite.
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
A topological space is an irreducible cut-point space if and only if X is homeomorphic to the Khalimsky line.
