
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the unit interval is the
closed interval , 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 ' (capital letter
). In addition to its role in
real analysis, the unit interval is used to study
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
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: , , and . However, the notation ' is most commonly reserved for the closed interval .
Properties
The unit interval is a
complete metric space,
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the
extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
. As a
topological space, it is
compact,
contractible,
path connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
and
locally path connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness a ...
. 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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
(every subset of the unit interval has a
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
and an
infimum).
Cardinality
The ''size'' or ''
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
'' of a set is the number of elements it contains.
The unit interval is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the
real numbers
. However, it has the same size as the whole set: the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
. Since the real numbers can be used to represent points along an
infinitely long line, this implies that a
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
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 1, as a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
of
volume 1, and even as an unbounded ''n''-dimensional
Euclidean space (see
Space filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
).
The number of elements (either real numbers or points) in all the above-mentioned sets is
uncountable, as it is strictly greater than the number of
natural numbers.
Generalizations
The interval , with length two, demarcated by the positive and negative units, occurs frequently, such as in the
range of the
trigonometric functions sine and cosine and the
hyperbolic function tanh. This interval may be used for the
domain of
inverse functions. For instance, when is restricted to 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 plays in homotopy theory. For example, in the theory of
quivers, 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 between quiver
homomorphisms analogous to the notion of homotopy between
continuous maps.
Fuzzy logic
In
logic, the unit interval can be interpreted as a generalization of the
Boolean domain , in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically,
negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
(NOT) is replaced with ;
conjunction (AND) is replaced with multiplication (); and
disjunction (OR) is defined, per
De Morgan's laws, as .
Interpreting these values as logical
truth values yields a
multi-valued logic, which forms the basis for
fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
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.
See also
{{wiktionary
*
Interval notation
* Unit
square,
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
,
circle,
hyperbola and
sphere
*
Unit impulse
*
Unit vector
References
* Robert G. Bartle, 1964, ''The Elements of Real Analysis'', John Wiley & Sons.
Sets of real numbers
1 (number)
Topology