Mathematics
In mathematics, the real line is the one-dimensional continuum of real numbers, usually denoted R or ℝ. It is the set of all points on a straight line extending infinitely in both directions, each point corresponding to a unique real number. The real line serves as the foundation for calculus, analysis, and topology, and it is the standard model for continuous quantities in science and engineering.
The real line is the set of real numbers together with its natural order and the usual metric. As an ordered set, it is complete in the sense that every nonempty bounded subset has a least upper bound (supremum)1. This completeness property distinguishes the real line from the rational line, which has gaps. The order topology on the real line is generated by open intervals (a, b), and this topology is homeomorphic to the usual Euclidean topology on ℝ2.
The real line is also a metric space with distance d(x, y) = |x − y|, and it is a complete metric space. It is connected, locally connected, and path-connected. As a topological space, it is second-countable and paracompact, making it a normal space3.
The real line is the prototypical example of a one-dimensional manifold. It is a smooth manifold with a global coordinate chart, and it is the universal cover of the circle S14. The real line is also the underlying space for the real projective line, which adds a point at infinity, and for the extended real line, which adds both +∞ and −∞5.
In analysis, the real line is the domain of real-valued functions of a real variable. The Lebesgue measure on the real line is the standard measure, and it is translation-invariant. The real line is also the setting for the Heine–Borel theorem, which characterizes compact subsets as closed and bounded6.
The real line has several surprising properties. For instance, it is not homeomorphic to the long line, a larger ordered continuum that is locally homeomorphic to ℝ but is not second-countable7. The real line is also the unique complete ordered field up to isomorphism, a result due to the work of Richard Dedekind and Georg Cantor1.
In point-set topology, the real line is a classic example of a space that is not compact but is σ-compact, being the union of countably many compact intervals. It is also a Polish space, as it is separable and completely metrizable. The real line can be embedded in the plane in many ways, and its image under a continuous injection is not necessarily a straight line, as shown by space-filling curves8.
The real line is the foundation for the real coordinate space ℝn, where it serves as the first coordinate axis. It is also the base for the real projective line and the Riemann sphere when extended with a point at infinity5. In physics, the real line models time in classical mechanics and one-dimensional space in quantum mechanics.
The real line is generalized to the surreal numbers, which form a proper class containing all real numbers and many more, and to the hyperreal numbers of nonstandard analysis, which include infinitesimals9. These extensions preserve the order but sacrifice the Archimedean property.
The real line is a foundational object in mathematics, appearing in nearly every branch of the discipline.
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