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The real numbers can be generalized and extended in several different directions: The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field. The affinely extended real number system adds two elements +∞ and −∞.
There is, however, exactly one infimum of the positive real numbers relative to the real numbers: , which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question.
The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M. Given the set of real numbers with the usual Euclidean metric and a subset defined as := (; /), then is a neighbourhood for the set of natural numbers, but is not a uniform neighbourhood of this set.
Also, the real numbers form an ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property: Every nonempty subset of that has an upper bound has a least upper bound that is also a real number.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
The indicator or characteristic function of a subset A of some set X maps elements of X to the codomain {,}. This mapping is surjective only when A is a non-empty proper subset of X . If A = X , {\displaystyle A=X,} then 1 A ≡ 1. {\displaystyle \mathbf {1} _{A}\equiv 1.}
On the real numbers , the usual less than relation < is a strict partial order. The same is also true of the usual greater than relation > on . By definition, every strict weak order is a strict partial order. The set of subsets of a given set (its power set) ordered by inclusion (see
The convex hull of a subset S of a real vector space is defined as the intersection of all convex sets that contain S. More concretely, the convex hull is the set of all convex combinations of points in S. In particular, this is a convex set. A (bounded) convex polytope is the convex hull of a finite subset of some Euclidean space R n.