Unit interval
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers 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, the unit interval is used to study homotopy theory in the field of topology.
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) (see Notations for intervals). However, the notation I is most commonly reserved for the closed interval [0,1].
Properties
The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space it is compact, contractible, path connected and locally path connected. The Hilbert cube 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 analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1.
The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).
Cardinality
Being homeomorpic to the extended real number line, the unit interval has the same cardinality as the real line: the cardinality of the continuum. Namely, the number of points in a line segment of length 1 strictly larger than the number of natural numbers, and hence it is "uncountable". However, it is equal to the number of points in a line of infinite length. Moreover, and even more surprisingly, the unit interval has the same cardinality as the unit square, or unit cube, or unit hypercube (see Space filling curve). Namely, the number of points in a line segment of length 1 is the same as the number of points in a square of area 1, a cube of volume 1, or a hypercube of volume 1.
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 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.
Fuzzy logic
In logic, the unit interval [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 (NOT) is replaced with conjunction (AND) is replaced with multiplication (), and disjunction (OR) is defined via De Morgan's law.
Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and 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.
References
- Robert G. Bartle, 1964, The Elements of Real Analysis, John Wiley & Sons.