In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
topology, a
topological space is said to be realcompact if it is completely
regular Hausdorff and every point of its
Stone–Čech compactification is real (meaning that the
quotient field at that point of the
ring 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 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 Janeiro.
...
). Realcompact spaces were introduced by .
Properties
*A space is realcompact if and only if it can be embedded
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
ally as a closed subset in some (not necessarily finite)
Cartesian power
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
of the reals, with the
product topology. 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 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
are realcompact.
*The (Hewitt) realcompactification υ''X'' of a topological space ''X'' consists of the real points of its
Stone–Čech compactification β''X''. A
topological space ''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 of realcompact spaces is dual to the category of rings of the form ''C''(''X'').
*In order that a
Hausdorff space ''X'' is compact it is necessary and sufficient that ''X'' is realcompact and pseudocompact (see Engelking, p. 153).
See also
*
Compact space
*
Paracompact space
*
Normal space
*
Pseudocompact space
*
Tychonoff space
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 ...
;
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 , location=Reading, Mass. .
Compactness (mathematics)
Properties of topological spaces