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Click any of our recommended sites and you'll visit a top real money casino for November Your guide to the best casinos, poker sites and sportsbooks: In addition to the above properties, the real line has no maximum or minimum element.

It also has a countable dense subset , namely the set of rational numbers. 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: 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 R.

This statement has been shown to be independent of the standard axiomatic system of set theory known as ZFC. The real line forms a metric space , with the distance function given by absolute difference:.

The metric tensor is clearly the 1-dimensional Euclidean metric. Since the n -dimensional Euclidean metric can be represented in matrix form as the n by n identity matrix, the metric on the real line is simply the 1 by 1 identity matrix, i.

The real line carries a standard 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 R are the same. As a topological space, the real line is homeomorphic to the open interval 0, 1.

The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 1-manifolds without 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. It is also path-connected , and is therefore 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 groups 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 R is a circle namely the real projective line , and the extra point can be thought of as an unsigned infinity.

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. The real line is a vector space over the field R of real numbers that is, over itself of dimension 1.

It has a standard inner product , making it a Euclidean space.

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The real line carries a canonical measure , namely the Lebesgue measure. This measure can be defined as the completion of a Borel measure defined on 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. Similarly, the algebra of quaternions.

In this way the real line consists of the fixed points of the conjugation. From Wikipedia, the free encyclopedia. This article covers advanced topics.

For basic topics, see Number line. This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations.

Please help to improve this article by introducing more precise citations. January Learn how and when to remove this template message. Absolute difference Cantor set Cantor—Dedekind axiom Completeness Construction Decidability of first-order theories Extended real number line Gregory number Irrational number Normal number Rational number Rational zeta series Real coordinate space Real line Tarski axiomatization Vitali set.

Retrieved from " https: Real numbers Topological spaces. Articles lacking in-text citations from January All articles lacking in-text citations.

Views Read Edit View history. This page was last edited on 21 August , at Both iPad and Fire tablet screen sizes will be able to accommodate multitable gaming, although you may have to squint a bit on your Samsung Galaxy or iPhone screen.

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