In mathematics, in the field of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

is said to be realcompact if it is completely regular Hausdorff and every point of its

Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Sto ...

is real (meaning that the

quotient field In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...

at that point of the

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

of real functions is the reals). Realcompact spaces have also been called Q-spaces, saturated spaces, functionally complete spaces, real-complete spaces, replete spaces and Hewitt–Nachbin spaces (named after

Edwin Hewitt Edwin Hewitt (January 20, 1920, Everett, Washington – June 21, 1999) was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage ze ...

and

Leopoldo Nachbin Leopoldo Nachbin (7 January 1922 – 3 April 1993) was a Jewish-Brazilian mathematician who dealt with topology, and harmonic analysis. Nachbin was born in Recife, and is best known for Nachbin's theorem. He died, aged 71, in Rio de Janei ...

). Realcompact spaces were introduced by .

Properties

*A space is realcompact if and only if it can be embedded homeomorphically as a closed subset in some (not necessarily finite) Cartesian power of the reals, with the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...

. Moreover, a (Hausdorff) space is realcompact if and only if it has the uniform topology and is

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...

for the

uniform structure In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...

generated by the continuous real-valued functions (Gillman, Jerison, p. 226). *For example

Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' s ...

s are realcompact; in particular all subsets of

\mathbb^n

are realcompact. *The (Hewitt) realcompactification υ''X'' of a topological space ''X'' consists of the real points of its

β''X''. A

''X'' is realcompact if and only if it coincides with its Hewitt realcompactification. *Write ''C''(''X'') for the ring of continuous real-valued functions on a topological space ''X''. If ''Y'' is a real compact space, then ring homomorphisms from ''C''(''Y'') to ''C''(''X'') correspond to continuous maps from ''X'' to ''Y''. In particular the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

of realcompact spaces is dual to the category of rings of the form ''C''(''X''). *In order that a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

''X'' is compact it is necessary and sufficient that ''X'' is realcompact and pseudocompact (see Engelking, p. 153).

References

Gillman, Leonard Leonard E. Gillman (January 8, 1917 – April 7, 2009) was an American mathematician, emeritus professor at the University of Texas at Austin. He was also an accomplished classical pianist. Biography Early life and education Gillman was born i ...

; Jerison, Meyer,
Rings of continuous functions
. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp. *. *. *{{Citation , last1=Willard, first1=Stephen, title=General Topology , year=1970 , publisher=

Addison-Wesley Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles throug ...

, location=Reading, Mass. . Compactness (mathematics) Properties of topological spaces

Properties

See also

References