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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. If a 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 used to determine whether two connected space are homeomorphic by counting the number of cut-points in each space. If they don't have the same number of cut-points, two spaces are not homeomorphic. This is a simple method to determine homeomorphism. A classic example is using cut-points to show that lines and circles are not homeomorphic.

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.

Definitions

Formal definitions:

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 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. Every point except for its end points are cut-points and the end-points {a,b} 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 {p,U,V} where p is a cut-point of X, U and V form the separation of X-{p}.
• Also can be written as X\{p}=U|V.

Theorems

Cut-points and homeomorphisms

• Cut-points are not necessarily preserved under continuous functions. For example: f: [0, 2π] → 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.

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-{a,b} isn't connected, K is homeomorphic to the unit circle.
• A topological space X is an irreducible cut-point space if and only if X is homeomorphic to the Khalimsky line.

Topological properties of cut-point spaces

• Let X be a connected space and x be a cut point in X such that X\{x}=A|B (A and B forms a separation of the resulting space). Then {x} is either open or closed. if {x} is open, A and B are closed. If {x} 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={ {2i-1,2i,2i+1}: i \in Z} union { {2i+1} : I \in Z} where B is a basis for a topology on Z. This is a cut-point space. Moreover, it's irreducible.

Theorem

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

Notes

• Hatcher, Allen, Notes on introductory point-set topology (PDF), p. 20-21

References

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

• Honari, B. and Bahrampour, Y. (1999), "Cut-point spaces", Proceedings of the American Mathematical Society, 127 (9): 2797 - 2803{{citation}}: CS1 maint: multiple names: authors list (link)