Unit interval in the context of Real line

In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite.

For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a].

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that every infinite sequence of points has limiting values. For example, the real line is not compact since the sequence of natural numbers has no real limiting value. The open interval (0,1) is not compact because it excludes the limiting values 0 and 1, whereas the closed interval [0,1] is compact. Similarly, the space of rational numbers ${\displaystyle \mathbb {Q} }$ is not compact, because every irrational number is the limit of the rational numbers that are lower than it. On the other hand, the extended real number line is compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence ⁠1/2⁠, ⁠4/5⁠, ⁠1/3⁠, ⁠5/6⁠, ⁠1/4⁠, ⁠6/7⁠, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering ${\displaystyle \mathbb {R} ^{1}}$ (the real number line), the sequence of points 0,  1,  2,  3, ... has no subsequence that converges to any real number.

In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point.

A loop may also be seen as a continuous map f from the pointed unit circle S into X, because S may be regarded as a quotient of I under the identification of 0 with 1.

In mathematics, fuzzy sets (also known as uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.At the same time, Salii (1965) defined a more general kind of structure called an "L-relation", which he studied in an abstract algebraic context; fuzzy relations are special cases of L-relations when L is the unit interval [0, 1].They are now used throughout fuzzy mathematics, having applications in areas such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978).

In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.

In mathematical logic, a Gödel logic, sometimes referred to as Dummett logic or Gödel–Dummett logic, is a member of a family of finite- or infinite-valued logics in which the sets of truth values V are closed subsets of the unit interval [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics. The concept is named after Kurt Gödel.

In 1959, Michael Dummett showed that infinite-valued propositional Gödel logic can be axiomatised by adding the axiom schema

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article by Tychonoff, "Über einen Funktionenraum".

Tychonoff's theorem is one of the most foundational results in general topology. The theorem is also valid for topological spaces based on fuzzy sets.