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==Advanced concepts== ===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]]. ===As a metric space=== [[File:Absolute difference.svg|thumb|The [[metric space|metric]] on the real line is [[absolute difference]].]] [[File:Epsilon Umgebung.svg|thumb|An {{math|''ε''}}-[[Ball (mathematics)|ball]] around a number {{math|''a''}}]] The real line forms a [[metric space]], with the [[distance function]] given by absolute difference: : <math>d(x, y) = |x - y|.</math> The [[metric tensor]] is clearly the 1-dimensional [[Euclidean metric]]. Since the {{mvar|n}}-dimensional Euclidean metric can be represented in matrix form as the {{mvar|n}}-by-{{mvar|n}} identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1. If {{math|''p'' ∈ '''R'''}} and {{math|''ε'' > 0}}, then the {{mvar|ε}}-[[Ball (mathematics)|ball]] in {{math|'''R'''}} centered at {{mvar|p}} is simply the open [[Interval (mathematics)|interval]] {{math|(''p'' − ''ε'', ''p'' + ''ε'')}}. This real line has several important properties as a metric space: * The real line is a [[complete metric space]], in the sense that any [[Cauchy sequence]] of points converges. * The real line is [[path-connected]] and is one of the simplest examples of a [[geodesic metric space]]. * The [[Hausdorff dimension]] of the real line is equal to one. ===As a topological space=== [[Image:Real Projective Line (RP1).png|thumb|The real line can be [[Compactification (mathematics)|compactified]] by adding a [[point at infinity]].]] The real line carries a standard [[topological space|topology]], which can be introduced in two different, equivalent ways. First, since the real numbers are [[totally ordered]], they carry an [[order topology]]. Second, the real numbers inherit a [[metric topology]] from the metric defined above. The order topology and metric topology on {{math|'''R'''}} are the same. As a topological space, the real line is [[homeomorphic]] to the open interval {{math|(0, 1)}}. The real line is trivially a [[topological manifold]] of [[dimension]] {{Num|1}}. Up to homeomorphism, it is one of only two different connected 1-manifolds without [[manifold with boundary|boundary]], the other being the [[circle]]. It also has a standard differentiable structure on it, making it a [[differentiable manifold]]. (Up to [[diffeomorphism]], there is only one differentiable structure that the topological space supports.) The real line is a [[locally compact space]] and a [[paracompact space]], as well as [[second-countable]] and [[normal space|normal]]. It is also [[path-connected]], and is therefore [[connected space|connected]] as well, though it can be disconnected by removing any one point. The real line is also [[contractible]], and as such all of its [[homotopy group]]s and [[reduced homology]] groups are zero. As a locally compact space, the real line can be compactified in several different ways. The [[one-point compactification]] of {{math|'''R'''}} is a circle (namely, the [[real projective line]]), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two [[End (topology)|ends]], and the resulting end compactification is the [[extended real number line]] {{math|[−∞, +∞]}}. There is also the [[Stone–Čech compactification]] of the real line, which involves adding an infinite number of additional points. In some contexts, it is helpful to place other topologies on the set of real numbers, such as the [[lower limit topology]] or the [[Zariski topology]]. For the real numbers, the latter is the same as the [[finite complement topology]]. ===As a vector space=== [[File:Bijection between vectors and points on number line.svg|thumb|The bijection between points on the real line and vectors]] The real line is a [[vector space]] over the [[field (mathematics)|field]] {{math|'''R'''}} of real numbers (that is, over itself) of [[dimension]] {{Num|1}}. It has the usual multiplication as an [[inner product]], making it a [[Euclidean vector space]]. The [[Norm (mathematics)|norm]] defined by this inner product is simply the [[absolute value]]. ===As a measure space=== The real line carries a canonical [[Measure (mathematics)|measure]], namely the [[Lebesgue measure]]. This measure can be defined as the [[Complete measure|completion]] of a [[Borel measure]] defined on {{math|'''R'''}}, where the measure of any interval is the length of the interval. Lebesgue measure on the real line is one of the simplest examples of a [[Haar measure]] on a [[locally compact group]]. ===In real algebras=== When ''A'' is a unital [[algebra over a field|real algebra]], the products of real numbers with 1 is a real line within the algebra. For example, in the [[complex plane]] ''z'' = ''x'' + i''y'', the subspace {''z'' : ''y'' = 0} is a real line. Similarly, the algebra of [[quaternion]]s :''q'' = ''w'' + ''x'' i + ''y'' j + ''z'' k has a real line in the subspace {''q'' : ''x'' = ''y'' = ''z'' = 0}. When the real algebra is a [[direct sum of modules|direct sum]] <math>A = R \oplus V,</math> then a '''conjugation''' on ''A'' is introduced by the mapping <math>v \to -v</math> of subspace ''V''. In this way the real line consists of the [[fixed point (mathematics)|fixed point]]s of the conjugation. For a dimension ''n'', the [[square matrices]] form a [[ring (mathematics)|ring]] that has a real line in the form of real products with the [[identity matrix]] in the ring.
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