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Unit interval
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Unit interval
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The unit interval as a subset of the real line

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 (capital letter 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). However, the notation I is most commonly reserved for the closed interval [0,1].

Properties

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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

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Main article: Cardinality of the continuum

The size or cardinality of a set is the number of elements it contains.

The unit interval is a subset of the real numbers . However, it has the same size as the whole set: the cardinality of the continuum. Since the real numbers can be used to represent points along an infinitely long line, this implies that a 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 1, as a cube of volume 1, and even as an unbounded n-dimensional Euclidean space (see Space filling curve).

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.

Orientation

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The unit interval is a curve. The open interval (0,1) is a subset of the positive real numbers and inherits an orientation from them. The orientation is reversed when the interval is entered from 1, such as in the integral used to define 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 yielding negative area over the unit interval due to reversed orientation there.

Generalizations

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The interval [-1,1], 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 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 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

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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 1 − x; conjunction (AND) is replaced with multiplication (xy); and disjunction (OR) is defined, per De Morgan's laws, as 1 − (1 − x)(1 − y).

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.

See also

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Look up unit interval in Wiktionary, the free dictionary.

References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Unit interval

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from Grokipedia
In , the unit interval is the closed interval [0,1][0,1] consisting of all real numbers xx such that 0≤x≤10 \leq x \leq 1. This set is a fundamental object in , , and related fields, serving as a for studying properties of intervals on the real line. Variants include the open unit interval (0,1)(0,1), which excludes the endpoints, and half-open forms like [0,1)[0,1) or $(0,1]$. The closed unit interval [0,1][0,1] exhibits key topological properties: it is compact, meaning every open cover has a finite subcover, and connected, meaning it cannot be expressed as the union of two disjoint non-empty open sets. These attributes make it a example in point-set , where it is used to define path-connectedness—a space is path-connected if any two points can be joined by a continuous path, which is a continuous map from [0,1][0,1] to the . Furthermore, continuous images of [0,1][0,1] characterize compact, connected, locally connected metric s, highlighting its role in embedding theorems and the study of continua. In , the unit interval often models the uniform distribution, where outcomes are equally likely across [0,1][0,1], providing a standard with as the . This setup underpins in and simulations, as pseudorandom generators typically produce values in [0,1][0,1] that approximate this uniform distribution. In , [0,1][0,1] is central to integration , fixed-point theorems like Brouwer's (which guarantees a fixed point for continuous self-maps of the unit interval or ball), and the of fractals such as the by iterative removal of middle thirds.

Definition and Fundamentals

Definition

The unit interval, often denoted by II, is the closed subset of the real numbers consisting of all points between 0 and 1, inclusive of the endpoints:
I={x∈R∣0≤x≤1}.I = \{ x \in \mathbb{R} \mid 0 \leq x \leq 1 \}.
This set includes the boundary points 0 and 1, forming a bounded segment on the real line. Visually, it represents a straight line segment starting at 0 and ending at 1, serving as a fundamental one-dimensional object in analysis and topology. The concept of the unit interval builds on Georg Cantor's late 19th-century work in , where he examined intervals of real numbers, building on his 1874 proof of the uncountability of the reals, and in 1877 demonstrated that the unit interval has the same as the unit cube in any finite number of dimensions. The term and its explicit use in modern mathematical contexts emerged in early 20th-century analysis, with significant contributions from , who in his 1914 book Grundzüge der Mengenlehre analyzed intervals like [0,1] in the development of axiomatic and . Although variants such as the open interval (0,1)(0,1), which excludes the endpoints, or half-open intervals like [0,1)[0,1), are used in certain contexts, the closed unit interval [0,1] is the conventional choice, especially for its topological .

Notation and Conventions

The unit interval is primarily denoted using the closed interval notation [0,1][0,1], where the square brackets indicate inclusion of the endpoints 0 and 1, following the standard interval notation for bounded closed intervals on the real line. This notation emphasizes the set {x∈R∣0≤x≤1}\{ x \in \mathbb{R} \mid 0 \leq x \leq 1 \}. In many mathematical texts, particularly in and , it is also commonly abbreviated as the single capital letter II, defined explicitly as I=[0,1]I = [0,1]. Variants of the unit interval employ different endpoint inclusions to suit analytical needs: the open unit interval is denoted (0,1)={x∈R∣0<x<1}(0,1) = \{ x \in \mathbb{R} \mid 0 < x < 1 \}, while half-open forms include [0,1)={x∈R∣0≤x<1}[0,1) = \{ x \in \mathbb{R} \mid 0 \leq x < 1 \} and $(0,1] = { x \in \mathbb{R} \mid 0 < x \leq 1 }.[](https://mathworld.wolfram.com/UnitInterval.html)Theclosedform.[](https://mathworld.wolfram.com/UnitInterval.html) The closed form [0,1]ispreferredintopologicaldiscussionsduetoitscompactnessandconnectednessasasubspaceofis preferred in topological discussions due to its compactness and connectedness as a subspace of\mathbb{R},whereastheopen, whereas the open (0,1)$ is often favored in real analysis and measure theory to focus on interior points without boundary complications. Typographical conventions for the unit interval in print typically render it in italics or boldface to distinguish it as a mathematical object, ensuring clarity in dense prose. In digital typesetting with LaTeX, the standard command is $[0,1]$ for inline usage or $$ [0,1] $$ for display, which automatically handles spacing and font styling within math mode. These practices promote consistent representation across diverse mathematical literature.

Properties

Topological Properties

The unit interval [0,1][0,1], equipped with the subspace topology inherited from the real line R\mathbb{R} under the standard topology, exhibits several fundamental topological properties that underscore its role as a prototypical compact space. As a closed and bounded subset of R\mathbb{R}, it satisfies the conditions of the Heine-Borel theorem, which states that a subset of Rn\mathbb{R}^n is compact if and only if it is closed and bounded. Specifically, [0,1][0,1] is closed because its complement in R\mathbb{R} is the union of the open intervals (−∞,0)(-\infty, 0) and (1,∞)(1, \infty), and it is bounded since all its points lie within the open ball of radius 1 centered at the origin in R\mathbb{R}. Consequently, every open cover of [0,1][0,1] admits a finite subcover, ensuring that continuous images of [0,1][0,1] are compact and that it supports key theorems in analysis and topology. The space [0,1][0,1] is also connected, meaning it cannot be expressed as the union of two disjoint nonempty open sets. This follows from the fact that connected subspaces of R\mathbb{R} are precisely the intervals, and [0,1][0,1] is such an interval. Moreover, [0,1][0,1] is path-connected: for any two points x,y∈[0,1]x, y \in [0,1] with x<yx < y, the straight-line path γ(t)=x+t(y−x)\gamma(t) = x + t(y - x) for t∈[0,1]t \in [0,1] is a continuous map from [0,1][0,1] to [0,1][0,1] connecting them. Path-connectedness implies connectedness, reinforcing the indivisibility of [0,1][0,1] in the topological sense. The metric structure on [0,1][0,1] is induced by the Euclidean metric on R\mathbb{R}, defined by d(x,y)=∣x−y∣d(x,y) = |x - y| for x,y∈[0,1]x, y \in [0,1]. This makes [0,1][0,1] a complete metric space, as every Cauchy sequence in [0,1][0,1] converges to a point within it, inheriting completeness from the closed embedding in R\mathbb{R}. It is also totally bounded, coverable by finitely many open balls of any positive ϵ>0\epsilon > 0 (for instance, by ⌈1/ϵ⌉\lceil 1/\epsilon \rceil balls of ϵ\epsilon), and has 1, the supremum of distances between its points, attained at the endpoints 0 and 1. Regarding homeomorphisms, [0,1][0,1] is homeomorphic to any closed bounded interval [a,b][a,b] with a<ba < b via the affine map f(x)=a+(b−a)xf(x) = a + (b-a)x, which is continuous, bijective, and has a continuous inverse. However, [0,1][0,1] is not homeomorphic to the open unit interval (0,1)(0,1), as removing an interior point from [0,1][0,1] disconnects it into two components, whereas removing any point from (0,1)(0,1) leaves it connected. Locally, [0,1][0,1] is compact and metrizable, with every point possessing a compact neighborhood, such as a closed subinterval contained within it. Its topology has a basis consisting of sets of the form (c,d)∩[0,1](c,d) \cap [0,1] where c<dc < d are real numbers, which includes half-open intervals at the endpoints like [0,d)[0,d) for 0<d≤10 < d \leq 1 and (c,1](c,1] for 0≤c<10 \leq c < 1. This basis is countable when restricted to rational endpoints, confirming that [0,1][0,1] is second countable.

Order Properties

The unit interval [0,1][0,1], equipped with the standard order ≤\leq inherited from the real numbers R\mathbb{R}, is a totally ordered set: for any x,y∈[0,1]x,y \in [0,1], either x≤yx \leq y or y≤xy \leq x, with $0 serving as the minimum element and &#36;1 as the maximum element. This order is linear and antisymmetric, ensuring a unique total ranking of its elements without incomparabilities. The order on [0,1][0,1] is dense, such that between any two distinct points a<ba < b in the interval, there exists at least one cc with a<c<ba < c < b; this property follows from the density of both rational and irrational numbers within the reals, restricted to the bounded segment [0,1][0,1]. Density implies that the order has no "gaps," allowing for infinite subdivision while maintaining the total ordering. Under the lattice operations defined by the minimum (meet, ∧\wedge) and maximum (join, ∨\vee), [0,1][0,1] forms a complete lattice: for any subset S⊆[0,1]S \subseteq [0,1], the infimum inf⁡S=⋀S\inf S = \bigwedge S and supremum sup⁡S=⋁S\sup S = \bigvee S exist and belong to [0,1][0,1], bounded by the global minimum $0 and maximum &#36;1. This structure supports the computation of meets and joins for arbitrary collections, reflecting the completeness of the underlying real order. As an oriented interval, [0,1][0,1] possesses a natural direction from $0 to &#36;1, which is preserved by monotone functions: non-decreasing maps f:[0,1]→[0,1]f: [0,1] \to [0,1] maintain the order relations, ensuring x≤yx \leq y implies f(x)≤f(y)f(x) \leq f(y). Such functions respect the interval's inherent progression along the order. The unit interval is convex in R\mathbb{R}, meaning that for any x,y∈[0,1]x,y \in [0,1] and λ∈[0,1]\lambda \in [0,1], the convex combination λx+(1−λ)y\lambda x + (1-\lambda)y lies entirely within [0,1][0,1], forming the line segment between xx and yy. This convexity underscores the interval's role as a connected segment under the linear order.

Cardinality

The unit interval [0,1][0,1] is uncountable, a fact established by Georg Cantor's diagonal argument from 1891. To see this, suppose for contradiction that there exists a countable enumeration {xn}n=1∞\{x_n\}_{n=1}^\infty of all elements in [0,1][0,1], where each xnx_n has a decimal expansion xn=0.dn1dn2dn3…x_n = 0.d_{n1}d_{n2}d_{n3}\dots with digits dni∈{0,1,…,9}d_{ni} \in \{0,1,\dots,9\}. Construct a number x=0.d1d2d3⋯∈[0,1]x = 0.d_1 d_2 d_3 \dots \in [0,1] by setting dk=4d_k = 4 if the kk-th digit of xkx_k is 9, and dk=9d_k = 9 otherwise. Then xx differs from xkx_k in the kk-th decimal place for every kk, so xx is not in the enumeration, yielding a contradiction. This argument applies to a subset of [0,1][0,1] with representations using only digits 4 and 9 to avoid non-unique expansions, but extends to the full interval, proving uncountability. The cardinality of [0,1][0,1], denoted ∣[0,1]∣|[0,1]|, equals the cardinality of the continuum c=2ℵ0\mathfrak{c} = 2^{\aleph_0}. This follows from the existence of a bijection between (0,1)(0,1) and R\mathbb{R}, composed as x↦π(x−1/2)x \mapsto \pi(x - 1/2) followed by the tangent function, yielding f(x)=tan⁡(π(x−1/2))f(x) = \tan(\pi(x - 1/2)), which maps (0,1)(0,1) bijectively onto R\mathbb{R}; the endpoints 0 and 1 add only two elements, preserving cardinality via the Schröder–Bernstein theorem. Moreover, ∣[0,1]∣=∣P(N)∣|[0,1]| = |\mathcal{P}(\mathbb{N})|, the cardinality of the power set of the natural numbers, via binary expansions: each x∈[0,1]x \in [0,1] corresponds to a sequence (b1,b2,… )(b_1, b_2, \dots) where x=∑n=1∞bn/2nx = \sum_{n=1}^\infty b_n / 2^n and bn∈{0,1}b_n \in \{0,1\}, identifying the subset {n∣bn=1}⊆N\{n \mid b_n = 1\} \subseteq \mathbb{N}; non-uniqueness for dyadic rationals affects only countably many points and does not alter the overall cardinality. The continuum hypothesis (CH) asserts that c=ℵ1\mathfrak{c} = \aleph_1, meaning no infinite cardinal lies strictly between ℵ0\aleph_0 (the cardinality of N\mathbb{N}) and 2ℵ02^{\aleph_0}. CH is independent of the Zermelo–Fraenkel set theory with the axiom of choice (ZFC): Kurt Gödel proved in 1938 that ZFC is consistent with CH (and the generalized continuum hypothesis) by constructing the inner model LL of constructible sets satisfying these axioms, assuming ZFC's consistency. Paul Cohen showed in 1963 that ZFC is also consistent with the negation of CH using the forcing technique to build models where 2ℵ0>ℵ12^{\aleph_0} > \aleph_1. Thus, neither CH nor its negation can be derived from ZFC alone. Although uncountable, [0,1][0,1] contains a countable dense subset, namely Q∩[0,1]\mathbb{Q} \cap [0,1]. The set Q\mathbb{Q} is countable as the union over positive integers qq of the finite sets of fractions p/qp/q in lowest terms with p∈Zp \in \mathbb{Z}, so any subset is countable; density follows from the density of the rational numbers in the real numbers, which implies that every non-empty open subinterval of [0,1] contains a rational number. This countable dense subset highlights the distinction between combinatorial size and topological density in [0,1][0,1].

Measure and Integration

Lebesgue Measure

The Lebesgue measure μ\mu on the unit interval [0,1][0,1] is defined such that μ([0,1])=1\mu([0,1]) = 1, providing a complete, translation-invariant measure on the Borel σ\sigma-algebra that extends the intuitive notion of to more general sets. The μ∗(E)\mu^*(E) for any subset E⊆[0,1]E \subseteq [0,1] is given by the infimum of the sums of lengths of countable open covers of EE, while the inner measure uses suprema over measures of compact subsets; a set is Lebesgue measurable if these coincide. This measure is σ\sigma-additive on the Borel σ\sigma-algebra, ensuring countable unions of disjoint measurable sets have measures summing to the measure of the union. The Borel σ\sigma-algebra B([0,1])\mathcal{B}([0,1]) consists of all sets generated by the open intervals within [0,1][0,1], and includes all open and closed subsets of the unit interval as Borel measurable sets. Every open set in [0,1][0,1] is a countable union of such intervals with rational endpoints, confirming the generative role of intervals. Although the Lebesgue measure covers a rich class of sets, not all subsets of [0,1][0,1] are measurable; the Vitali set, constructed by partitioning [0,1][0,1] into equivalence classes under rational translations and selecting one representative from each using the axiom of choice, exemplifies a non-Lebesgue measurable subset. This construction yields a set whose measure cannot be consistently defined, as its rational translates are disjoint and cover [0,1][0,1] up to measure zero, yet their total measure would contradict the unit length if assigned a value. The unit interval serves as a foundational domain for Lebesgue integration, where the space L1([0,1])L^1([0,1]) comprises equivalence classes of measurable functions f:[0,1]→Rf: [0,1] \to \mathbb{R} with finite ∫[0,1]∣f∣ dμ<∞\int_{[0,1]} |f| \, d\mu < \infty, equipped with the norm ∥f∥1=∫[0,1]∣f∣ dμ\|f\|_1 = \int_{[0,1]} |f| \, d\mu. The Lebesgue integral extends Riemann integration to broader classes of functions, including those discontinuous on sets of measure zero. Compactness of [0,1][0,1] implies uniform continuity for continuous functions, facilitating their measurability and integrability. Lebesgue measure on [0,1][0,1] exhibits translation invariance for shifts by t∈Rt \in \mathbb{R}, where μ(E+tmod  1)=μ(E)\mu(E + t \mod 1) = \mu(E) for measurable EE, though the interval's lack of group structure under addition modulo 1 limits full invariance compared to R\mathbb{R}.

Riemann Integration

The Riemann integral provides a foundational method for computing the integral of a function defined on the unit interval [0,1]. For a bounded function f:[0,1]→Rf: [0,1] \to \mathbb{R}, the integral ∫01f(x) dx\int_0^1 f(x) \, dx is defined as the limit of Riemann sums ∑f(xi∗)Δxi\sum f(x_i^*) \Delta x_i, where the interval is partitioned into subintervals of lengths Δxi=xi−xi−1\Delta x_i = x_i - x_{i-1} and xi∗x_i^* is a point in each subinterval [xi−1,xi][x_{i-1}, x_i], taken as the norm of the partition approaches zero. If ff is continuous on the compact interval [0,1], it is uniformly continuous, ensuring the limit exists and is independent of the choice of points xi∗x_i^*. This guarantees that every continuous function on [0,1] is Riemann integrable. An equivalent formulation, known as the Darboux integral, defines integrability through upper and lower sums. For a partition P={x0=0,x1,…,xn=1}P = \{x_0 = 0, x_1, \dots, x_n = 1\}, the upper sum is U(f,P)=∑MiΔxiU(f,P) = \sum M_i \Delta x_i where Mi=sup⁡[xi−1,xi]fM_i = \sup_{[x_{i-1},x_i]} f, and the lower sum is L(f,P)=∑miΔxiL(f,P) = \sum m_i \Delta x_i where mi=inf⁡[xi−1,xi]fm_i = \inf_{[x_{i-1},x_i]} f. The function ff is Riemann integrable if the upper integral inf⁡PU(f,P)\inf_P U(f,P) equals the lower integral sup⁡PL(f,P)\sup_P L(f,P), with the common value being the integral. A criterion for integrability states that ff is integrable if and only if for every ϵ>0\epsilon > 0, there exists a partition PP such that U(f,P)−L(f,P)<ϵU(f,P) - L(f,P) < \epsilon, which relates to the oscillation of ff being controlled on fine partitions. This approach emphasizes the convergence of sums without tagged points, aligning closely with the Riemann sum definition. The connects differentiation and integration on [0,1]. If ff is Riemann integrable on [0,1] and F(x)=∫0xf(t) dtF(x) = \int_0^x f(t) \, dt, then FF is continuous on [0,1] and differentiable on (0,1) with F′(x)=f(x)F'(x) = f(x) ; moreover, if FF is an of ff (i.e., F′(x)=f(x)F'(x) = f(x) for all x∈[0,1]x \in [0,1]), then ∫01f(x) dx=F(1)−F(0)\int_0^1 f(x) \, dx = F(1) - F(0). For example, taking f(x)=xf(x) = x, the is F(x)=12x2F(x) = \frac{1}{2}x^2, so ∫01x dx=F(1)−F(0)=12\int_0^1 x \, dx = F(1) - F(0) = \frac{1}{2}. This theorem underpins the evaluation of definite integrals via antiderivatives on the unit interval. Improper Riemann integrals extend the definition to functions unbounded near endpoints. For ff continuous on (0,1](0,1] but unbounded at 0, the is ∫01f(x) dx=lim⁡ϵ→0+∫ϵ1f(x) dx\int_0^1 f(x) \, dx = \lim_{\epsilon \to 0^+} \int_\epsilon^1 f(x) \, dx, provided the limit exists. For instance, f(x)=1/xf(x) = 1/\sqrt{x}
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