Cut point explained

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

• A closed interval [a,b] 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 (a,b) 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: [0, 2{{pi}}] → R², given by f(x) = (cos x, sin x). Every point of the interval (except the two endpoints) is a cut-point, but f(x) 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 (compact connected Hausdorff space) 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.

Topological properties of cut-point spaces

• 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.

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

Z be the set of the integers and B = \{ \{ 2 i -1 , 2 i, 2 i +1 \} : i \in Z \} \cup \{ \{ 2 i +1 \} : i \in Z \} where B is a basis for a topology on Z . The Khalimsky line is the set Z endowed with this topology. It's a cut-point space. Moreover, it's irreducible.

Theorem

• A topological space

X is an irreducible cut-point space if and only if X is homeomorphic to the Khalimsky line.

See also

Cut point (graph theory)

References

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