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). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... 2-dimensional renderings (ie. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... This article discusses orientability and orientation on surfaces and manifolds. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...

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 is most commonly reserved for the closed interval [0,1].

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 {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 mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed. ... The two bold paths shown above are homotopic relative to their endpoints. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

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

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0     1
   I