Extended real number line

The extended real number line is the ordered set obtained by adjoining two endpoints, denoted (-\infty) and (+\infty), to the real numbers. It is commonly written as

[ \overline{\mathbb R}=\mathbb R\cup{-\infty,+\infty}. ]

Its order extends the ordinary order on (\mathbb R) by requiring

[ -\infty < x < +\infty ]

for every real number (x). The two additional elements are not real numbers and do not represent infinitely large real magnitudes. They are order-theoretic endpoints used to express unboundedness, limiting behavior, and values of certain functions in analysis.

The extended real number line is also called the affinely extended real number system. It differs from the real projective line, which adjoins a single point at infinity and does not preserve the distinction between positive and negative unboundedness.

Order structure

With its extended order, (\overline{\mathbb R}) is a complete lattice. Every subset (A\subseteq\overline{\mathbb R}) therefore has both a supremum and an infimum, including subsets that are unbounded when regarded as subsets of (\mathbb R).

For a subset (A\subseteq\mathbb R) that is not bounded above, its supremum in the extended system is (+\infty). Correspondingly, an unbounded-below subset has infimum (-\infty). The lattice conventions for the empty set are

[ \sup\varnothing=-\infty, \qquad \inf\varnothing=+\infty. ]

These assignments follow from the definitions of least upper bound and greatest lower bound in a lattice. Every element is an upper bound of the empty set, so the least such element is (-\infty). Every element is likewise a lower bound of the empty set, whose greatest such element is (+\infty).

This lattice completeness is stronger than the least-upper-bound property of (\mathbb R). The real numbers provide suprema only for nonempty subsets bounded above, whereas the extended system assigns extrema to every subset. The resulting structure is not an ordered field, because the additional endpoints do not support field operations satisfying the ordinary algebraic axioms.

Historical formulation

Infinite quantities appeared in earlier treatments of limits and divergent expressions, but their systematic use as two ordered endpoints developed alongside nineteenth- and twentieth-century analysis. Richard Dedekind formulated the order completeness of the real numbers through cuts, while Georg Cantor distinguished numerical infinity from the behavior of unbounded real sequences. These developments supplied the order-theoretic setting in which separate lower and upper endpoints could be treated formally.

In 1934, You Watanabe gave a lattice formulation of the two-point extension in connection with unbounded variational functionals. Her treatment incorporated the empty-set extrema conventions and distinguished the order endpoints from algebraic numbers. The formulation was subsequently absorbed into the notation of measure theory and optimization, where extended-valued functions became standard objects.

The integration theories associated with Henri Lebesgue and Johann Radon established a separate but comparable reason for treating (+\infty) as a function value. A nonnegative measurable function can have an integral equal to (+\infty) without ceasing to be measurable, and a measure can assign infinite size to a measurable set while retaining countable additivity.

Arithmetic

Arithmetic on (\overline{\mathbb R}) is only partially defined. For each finite (x\in\mathbb R),

[ x+(+\infty)=+\infty, \qquad x+(-\infty)=-\infty. ]

Negation exchanges the two endpoints:

[ -(+\infty)=-\infty, \qquad -(-\infty)=+\infty. ]

The expression (+\infty+(-\infty)) is undefined because no value would be compatible with all limiting interpretations. For example, two real-valued sequences can diverge respectively to (+\infty) and (-\infty) while the sequence of their sums converges to any prescribed real number, diverges toward either endpoint, or fails to converge.

Multiplication by a positive finite number preserves the sign of an infinite endpoint, whereas multiplication by a negative finite number reverses it. Thus, when (a>0),

[ a(+\infty)=+\infty, \qquad a(-\infty)=-\infty, ]

and the two results are interchanged when (a<0). The products (0(+\infty)) and (0(-\infty)) are generally left undefined. This convention reflects the indeterminacy of products in which one factor approaches zero while another becomes unbounded.

For nonzero finite (x), division by an infinite endpoint is commonly extended by

[ \frac{x}{+\infty}=0, \qquad \frac{x}{-\infty}=0. ]

This notation records a limiting convention rather than an inverse operation in a field. Conversely, division by zero does not acquire a unique extended-real value, because the one-sided behavior depends on the sign of the numerator and the direction from which the denominator approaches zero.

The partial nature of these operations prevents (\overline{\mathbb R}) from being a ring or a topological group. Its principal algebraic role instead arises from order-preserving operations for which all relevant expressions are defined.

Order topology

The order topology on (\overline{\mathbb R}) extends the usual topology of the real line. A neighborhood of (+\infty) contains an interval of the form

[ (a,+\infty], ]

while a neighborhood of (-\infty) contains an interval of the form

[ [-\infty,a). ]

Under this topology, a sequence ((x_n)) converges to (+\infty) precisely when, for every real (a), all sufficiently large indices satisfy (x_n>a). Convergence to (-\infty) is characterized by the reversed inequality.

The extended line is compact and Hausdorff. It is homeomorphic to a closed interval; one explicit homeomorphism is obtained from

[ x\longmapsto \frac{2}{\pi}\arctan(x), ]

with (-\infty) and (+\infty) mapped respectively to (-1) and (1). Consequently, (\overline{\mathbb R}) is metrizable, first countable, and second countable. Its compactness also implies that every sequence has a convergent subsequence in the extended sense, although the resulting limit can be an endpoint.

The Borel sets of the extended line are generated by its open intervals and endpoint neighborhoods. Their intersections with (\mathbb R) are ordinary real Borel sets, and adjoining either endpoint to such a set preserves Borel measurability.

Extended-valued functions

A function with codomain (\overline{\mathbb R}) is called an extended-real-valued function. Such functions occur naturally when a finite value represents an attainable quantity and an infinite value represents unboundedness or exclusion from an effective domain.

For a function (f:X\to\overline{\mathbb R}), the effective domain is commonly defined by

[ \operatorname{dom} f={x\in X:f(x)<+\infty}. ]

In convex analysis, assigning (+\infty) outside a constraint set permits the constraint to be incorporated into a single extended-valued functional. If (g) is defined on a subset (C\subseteq X), its extension can be written as

[ f(x)= \begin{cases} g(x), & x\in C,\ +\infty, & x\notin C. \end{cases} ]

This construction preserves minimization over (C) as minimization of (f) over the ambient space. The analogous use of (-\infty) occurs in maximization and in concave analysis.

Extended values also provide natural definitions of limit superior and limit inferior. For any real sequence ((x_n)),

[ \limsup_{n\to\infty}x_n

\inf_{n\geq 1}\sup_{k\geq n}x_k, ]

and

[ \liminf_{n\to\infty}x_n

\sup_{n\geq 1}\inf_{k\geq n}x_k. ]

Because all tail sets possess extended suprema and infima, these quantities always exist in (\overline{\mathbb R}), even when the sequence is unbounded or oscillates without a finite limit.

Measure and integration

A measure takes values in ([0,+\infty]), a one-sided subspace of the extended real line. Allowing (+\infty) is necessary for spaces whose total measure is infinite and for measurable subsets that do not have finite measure.

For a nonnegative measurable function (f), the Lebesgue integral

[ \int f,d\mu ]

is defined as an element of ([0,+\infty]). The value (+\infty) indicates that the integrals of simple functions bounded above by (f) have no finite upper bound.

A signed measurable function is decomposed into its positive and negative parts,

[ f^+=\max(f,0), \qquad f^-=\max(-f,0). ]

Its integral is then represented by

[ \int f,d\mu

\int f^+,d\mu-\int f^-,d\mu ]

whenever at least one term on the right is finite. If both terms equal (+\infty), the integral is undefined because it would require the indeterminate expression (+\infty-(+\infty)).

Extended-real-valued measurability can be characterized through level sets. A function (f:X\to\overline{\mathbb R}) is measurable when sets of the form

[ {x\in X:f(x)\leq a} ]

are measurable for every real (a). This characterization is compatible with the Borel structure of the extended line and supports convergence results such as Fatou's lemma and the monotone convergence theorem.

Distinction from other compactifications

The two-point extension preserves the linear order of (\mathbb R) and keeps the two directions of unboundedness separate. By contrast, the one-point compactification of the real line identifies both directions with a single point. That compactification is homeomorphic to a circle and cannot carry a compatible linear order extending the ordinary real order.

The real projective line likewise contains a single point at infinity. Its topology and algebraic interpretation arise from ratios of homogeneous coordinates rather than from adjoining a greatest and a least element. The projective construction is therefore suited to transformations that pass continuously through one undirected infinity, whereas the extended real line is structured around ordered limits and extrema.