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 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 edit

Formal definitions edit

A line (closed interval) has infinitely many cut-points between two end points. A circle has no cut-point. Since they have different numbers of cut-points, lines are not homeomorphic to circles

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 edit

A closed interval [a,b] has infinitely many cut-points. All points 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 edit

A cutting of X is a set {p,U,V} where p is a cut-point of X, U and V form a separation of X-{p}.
Also can be written as X\{p}=U|V.

Theorems edit

Cut-points and homeomorphisms edit

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

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.

Topological properties of cut-point spaces edit

Let X be a connected space and x be a cut point in X such that X\{x}=A|B. 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 edit

Definitions edit

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

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

Theorem edit

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

References edit

Hatcher, Allen, Notes on introductory point-set topology, pp. 20–21
Honari, B.; Bahrampour, Y. (1999), "Cut-point spaces" (PDF), Proceedings of the American Mathematical Society, 127 (9): 2797–2803, doi:10.1090/s0002-9939-99-04839-x
Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6. (Originally published by Addison-Wesley Publishing Company, Inc. in 1970.)