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{{Short description|Closed interval [0,1] on the real number line}}

[[File:Unit-interval.svg|thumb|The unit interval as a [[Subset|subset]] of the [[Real line|real line]]]]

In [[Mathematics|mathematics]], the '''unit interval''' is the [[Interval (mathematics)|closed interval]] {{closed-closed|0,1}}, that is, the [[Set (mathematics)|set]] of all [[Real number|real number]]s that are greater than or equal to 0 and less than or equal to 1.  It is often denoted ''{{math|I}}'' (capital letter <big>{{mono|I}}</big>). In addition to its role in [[Real analysis|real analysis]], the unit interval is used to study [[Homotopy theory|homotopy theory]] in the field of [[Topology|topology]].

In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: {{open-closed|0,1}}, {{closed-open|0,1}}, and {{open-open|0,1}}.  However, the notation ''{{math|I}}'' is most commonly reserved for the closed interval {{closed-closed|0,1}}.

== Properties ==

The unit interval is a [[Complete metric space|complete metric space]], [[Homeomorphism|homeomorphic]] to the [[Extended real number line|extended real number line]]. As a [[Topological space|topological space]], it is [[Compact space|compact]], contractible, [[Connectedness|path connected]] and [[Locally connected space|locally path connected]]. The [[Hilbert cube]] is obtained by taking a topological product of countably many copies of the unit interval.

In [[Mathematical analysis|mathematical analysis]], the unit interval is a [[Physics:Dimension|one-dimensional]] analytical [[Manifold|manifold]] whose boundary consists of the two points 0 and 1. Its standard [[Orientability|orientation]] goes from 0 to 1.

The unit interval is a [[Total order|totally ordered set]] and a [[Complete lattice|complete lattice]] (every subset of the unit interval has a supremum and an infimum).

===Cardinality===
{{Main|Cardinality of the continuum}}

The ''size'' or ''[[Cardinality|cardinality]]'' of a set is the number of elements it contains.

The unit interval is a [[Subset|subset]] of the [[Real number|real number]]s <math>\mathbb{R}</math>. However, it has the same size as the whole set: the [[Cardinality of the continuum|cardinality of the continuum]]. Since the real numbers can be used to represent points along an [[Real line|infinitely long line]], this implies that a [[Line segment|line segment]] of length 1, which is a part of that line, has the same number of points as the whole line. Moreover, it has the same number of points as a square of [[Area|area]] 1, as a [[Cube|cube]] of [[Physics:Volume|volume]] 1, and even as an unbounded ''n''-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math> (see Space filling curve).

The number of elements (either real numbers or points) in all the above-mentioned sets is [[Uncountable set|uncountable]], as it is strictly greater than the number of [[Natural number|natural number]]s.

===Orientation===
The unit interval is a [[Curve|curve]]. The open interval (0,1) is a subset of the [[Positive real numbers|positive real numbers]] and inherits an orientation from them. The [[Curve orientation|orientation]] is reversed when the interval is entered from 1, such as in the integral <math>\int_1^x \frac{dt}{t} </math> used to define [[Natural logarithm|natural logarithm]] for ''x'' in the interval, thus yielding negative values for logarithm of such ''x''. In fact, this integral is evaluated as a [[Signed area|signed area]] yielding ''negative area'' over the unit interval due to reversed orientation there.

== Generalizations ==
The interval {{closed-closed|-1,1}}, with length two, demarcated by the positive and negative units, occurs frequently, such as in the [[Range of a function|range]] of the trigonometric functions sine and cosine and the [[Hyperbolic function|hyperbolic function]] tanh. This interval may be used for the [[Domain of a function|domain]] of [[Inverse function|inverse function]]s. For instance, when {{theta}} is restricted to {{closed-closed|&minus;π/2, π/2}} then <math>\sin\theta</math> is in this interval and arcsine is defined there.

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 {{closed-closed|0,1}} plays in homotopy theory.  For example, in the theory of [[Quiver (mathematics)|quiver]]s, the (analogue of the) unit interval is the graph whose vertex set is <math>\{0,1\}</math> 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]] between quiver [[Homomorphism|homomorphism]]s analogous to the notion of homotopy between continuous maps.

== Fuzzy logic ==
In [[Logic|logic]], the unit interval {{closed-closed|0,1}} can be interpreted as a generalization of the [[Boolean domain]] {0,1}, in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, [[Negation|negation]] (NOT) is replaced with {{math|1 − ''x''}}; [[Logical conjunction|conjunction]] (AND) is replaced with multiplication ({{math|''xy''}}); and [[Logical disjunction|disjunction]] (OR) is defined, per [[De Morgan's laws]], as {{math|1 − (1 − ''x'')(1 − ''y'')}}.

Interpreting these values as logical [[Philosophy:Truth value|truth value]]s yields a multi-valued logic, which forms the basis for [[Fuzzy logic|fuzzy logic]] and [[Probabilistic logic|probabilistic logic]]. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

==See also==
* Interval notation
* Unit [[Unit square|square]], [[Unit cube|cube]], [[Unit circle|circle]], [[Unit hyperbola|hyperbola]] and [[Unit sphere|sphere]]
* [[Unit impulse]]
* [[Unit vector]]

==References==
* Robert G. Bartle, 1964, ''The Elements of Real Analysis'', John Wiley & Sons.

[[Category:Sets of real numbers]]
[[Category:1 (number)]]
[[Category:Topology]]

{{Sourceattribution|Unit interval}}