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The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...
These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
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.
An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers. [1] If the infimum does not exist, one says often that the corresponding endpoint is .
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 ...
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine ...
Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of algebraic numbers is 0, even though the set is dense in . The Cantor set and the set of Liouville numbers are examples of uncountable sets that have Lebesgue measure 0.
After an infinite number of stages, infinite subsets become available, so that any real number a can be represented by { L a | R a}, where L a is the set of all dyadic rationals less than a and R a is the set of all dyadic rationals greater than a (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.