Unit interval

In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.

The unit interval is a complete metric space, homeomorphic to the extended real number line.

The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.

In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1.

Moreover, it has the same number of points as a square of area 1, as a cube of volume 1, and even as an unbounded n-dimensional Euclidean space

The open interval (0,1) is a subset of the positive real numbers and inherits an orientation from them.

The interval [-1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine and the hyperbolic function tanh.

Sometimes, the term "unit interval" is used to refer to objects that play a role in various branches of mathematics analogous to the role that [0,1] plays in homotopy theory.

For example, in the theory of quivers, the (analogue of the) unit interval is the graph whose vertex set is