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2. Completeness of the real numbers - Wikipedia

   en.wikipedia.org/wiki/Completeness_of_the_real...

   Dedekind completeness is the property that every Dedekind cut of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom. The rational number line Q is not Dedekind complete. An example is the Dedekind cut

3. Least-upper-bound property - Wikipedia

   en.wikipedia.org/wiki/Least-upper-bound_property

   In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .

4. Construction of the real numbers - Wikipedia

   en.wikipedia.org/wiki/Construction_of_the_real...

   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 ...

5. Completeness (order theory) - Wikipedia

   en.wikipedia.org/wiki/Completeness_(order_theory)

   In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions ...

6. Real number - Wikipedia

   en.wikipedia.org/wiki/Real_number

   In mathematics, a real number is a number that can be used to measure a continuous one-dimensional ... The completeness property of the reals is the basis on which ...

7. Real analysis - Wikipedia

   en.wikipedia.org/wiki/Real_analysis

   This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers ) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least upper bound property (see below).

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9. Infimum and supremum - Wikipedia

   en.wikipedia.org/wiki/Infimum_and_supremum

   The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness . If an ordered set S {\displaystyle S} has the property that every nonempty subset of S {\displaystyle S} having an upper bound also has a least upper ...