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. 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 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

Notations

Theorems

Cut-points and homeomorphisms

Cut-points and continua

Topological properties of cut-point spaces

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 ∈ Z} ∪ { {2i+1} : I ∈ 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

References

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