# Unit interval

*See also unit interval (telecommunications).*

In mathematics, the **unit interval** is the
interval [0,1], that is the set
of all real numbers `x` such that
zero is less than or equal to `x` and `x`
is less than or equal to one.
The unit interval plays a fundamental role in homotopy theory, a major branch of topology. It is a metric space, compact, contractible, path connected and locally path connected. As a topological space, it is homeomorphic to the extended real number line. The unit interval is a one-dimensional analytical manifold with boundary {0,1}, carrying a standard orientation from 0 to 1. As a subset of the real numbers, its Lebesgue measure is 1. It is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).

In the literature, the term "unit interval" is also sometimes applied to the other shapes that an interval from 0 to 1 could take, that is (0,1], [0,1), and (0,1). However, it's most commonly reserved for the closed interval [0,1], and Wikipedia follows this convention.

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 quiverss, the (analogue of the) unit interval is the graph whose vertex set is {0,1} and which contains a single edge `e` whose source is 0 and whose target is 1. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion of homotopy between continuous maps.

In all of its guises, the unit interval is almost always written `I`, and the following ASCII picture suffices in almost any context:

`*-->--*`

`0 1`

` I`