Unit interval

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Unit interval

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the unit interval is the closed interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
[0,1], that is, the set of all real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s that are greater than or equal to 0 and less than or equal to 1. It is often denoted I. In addition to its role in real analysis

Real analysis

Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the Set of real numbers. In particular, it deals with the analytic function properties of real function and sequences, including convergence and limit s of sequences of real numbers, the calculus of the real numbers, and continu...
, the unit interval is used to study homotopy theory in the field of topology

Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
.

In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1).

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Encyclopedia

In mathematics

Mathematics

Interval (mathematics)

Real number

Real analysis

Topology

Properties

The unit interval is a complete metric space, homeomorphic

Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
to the extended real number line

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +8 and −8 ....
. As a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
it is compact

Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
, contractible, path connected

Connectedness

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected....
and locally path connected

Locally connected space

In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets....
. The Hilbert cube

Hilbert cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology....
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

Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
analytical manifold

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
whose boundary consists of the two points 0 and 1. Its standard orientation

Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
goes from from 0 to 1.

The unit interval is a totally ordered set

Total order

In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X....
and a complete lattice

Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science....
(every subset of the unit interval has a supremum

Supremum

In mathematics, given a subset S of a partially ordered set T, the supremum of S, if it exists, is the greatest element of T that is greater than or equal to each element of S....
and an infimum

Infimum

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 elements of the subset....
).

Generalizations

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 quiver

Quiver (mathematics)

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertex are allowed. They are commonly used in representation theory: a representation, V, of a quiver assigns a vector space V to each vertex x of the quiver and a linear operator V to each arrow a....
s, the (analogue of the) unit interval is the graph whose vertex set is and which contains a single edge e whose source is 0 and whose target is 1. One can then define a notion of homotopy

Homotopy

In topology, two continuous function Function from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions....
between quiver homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
s analogous to the notion of homotopy between continuous

Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
maps.