Inmathematics, theunit interval is theclosed interval[0,1], that is, theset of allreal numbers that are greater than or equal to 0 and less than or equal to 1. It is often denotedI (capital letterI). In addition to its role inreal analysis, the unit interval is used to studyhomotopy theory in the field oftopology.
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). However, the notationI is most commonly reserved for the closed interval[0,1].
The unit interval is acomplete metric space,homeomorphic to theextended real number line. As atopological space, it iscompact,contractible,path connected andlocally path connected. TheHilbert cube is obtained by taking atopological product of countably many copies of the unit interval.
Inmathematical analysis, the unit interval is aone-dimensional analyticalmanifold whose boundary consists of the two points 0 and 1. Its standardorientation goes from 0 to 1.
The unit interval is atotally ordered set and acomplete lattice (every subset of the unit interval has asupremum and aninfimum).
Thesize orcardinality of a set is the number of elements it contains.
The unit interval is asubset of thereal numbers. However, it has the same size as the whole set: thecardinality of the continuum. Since the real numbers can be used to represent points along aninfinitely long line, this implies that aline 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 ofarea 1, as acube ofvolume 1, and even as an unboundedn-dimensionalEuclidean space (seeSpace filling curve).
The number of elements (either real numbers or points) in all the above-mentioned sets isuncountable, as it is strictly greater than the number ofnatural numbers.
The unit interval is acurve. The open interval (0,1) is a subset of thepositive real numbers and inherits an orientation from them. Theorientation is reversed when the interval is entered from 1, such as in the integral used to definenatural logarithm forx in the interval, thus yielding negative values for logarithm of suchx. In fact, this integral is evaluated as asigned area yieldingnegative area over the unit interval due to reversed orientation there.
The interval[-1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in therange of thetrigonometric functions sine and cosine and thehyperbolic function tanh. This interval may be used for thedomain ofinverse functions. For instance, when 𝜃 is restricted to[−π/2, π/2] then 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[0,1] plays in homotopy theory. For example, in the theory ofquivers, the (analogue of the) unit interval is the graph whose vertex set is and which contains a single edgee whose source is 0 and whose target is 1. One can then define a notion ofhomotopy between quiverhomomorphisms analogous to the notion of homotopy betweencontinuous maps.
Inlogic, the unit interval[0,1] can be interpreted as a generalization of theBoolean 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 (NOT) is replaced with1 −x;conjunction (AND) is replaced with multiplication (xy); anddisjunction (OR) is defined, perDe Morgan's laws, as1 − (1 −x)(1 −y).
Interpreting these values as logicaltruth values yields amulti-valued logic, which forms the basis forfuzzy logic andprobabilistic 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.
