Editing Number line (section)

===As a linear continuum===
[[File:Illustration of supremum.svg|thumb|Each set on the real number line has a supremum.]]
The real line is a [[linear continuum]] under the standard {{math|<}} ordering.  Specifically, the real line is [[linearly ordered]] by {{math|<}}, and this ordering is [[dense order|dense]] and has the [[least-upper-bound property]].

In addition to the above properties, the real line has no [[Greatest element|maximum]] or [[least element|minimum element]].  It also has a [[countable]] [[dense set|dense]] [[subset]], namely the set of [[rational number]]s.  It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is [[order-isomorphic]] to the real line.

The real line also satisfies the [[countable chain condition]]: every collection of mutually [[disjoint sets|disjoint]], [[nonempty]] open [[interval (mathematics)|interval]]s in {{math|'''R'''}} is countable.  In [[order theory]], the famous [[Suslin problem]] asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to {{math|'''R'''}}.  This statement has been shown to be [[independence (mathematical logic)|independent]] of the standard axiomatic system of [[set theory]] known as [[ZFC]].