In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a cut-point is a point of a connected space

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 such that its removal causes the resulting space to be disconnected. 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 in the characterization of topological continua

Continuum (topology)

In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...

, a class of spaces which combine the properties of compactness

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 and connectedness

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 and include many familiar spaces such as the unit interval

Unit interval

In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

, the circle, and the torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

.

Definition

A cut-point of a connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 T₁

T1 space

In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...

 topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 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 noncut-point.

Properties

• Cut-points are not necessarily preserved under continuous function
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
  s, (example: f: [0, 2π] → R², given by f(x) = (cos x, sin x)), but are preserved under homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  s. Therefore the circle is not homeomorphic to a line segment, as the circle has no cut-points, but every point of the interval (except the two endpoints) is a cut-point.
• Every compact connected Hausdorff space, with more than one point, has at least two noncut-points.
• Every compact connected metric space, with exactly two noncut-points is homeomorphic to the unit interval.