mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in the field of

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

is said to be realcompact if it is completely regular Hausdorff and it contains 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 property, universal map from a topological space ''X'' to a Compact space, compact Ha ...

that 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 fiel ...

at that point of the

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

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 z ...

and Leopoldo Nachbin). Realcompact spaces were introduced by .

Properties

*A space is realcompact if and only if it can be embedded

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

ally 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-seemin ...

. 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 t ...

for the

uniform structure In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...

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'' sub ...

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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

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 ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...

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

References

* Gillman, Leonard; 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 ...

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

Properties

See also

References