*Definition [#dc10508f]
Let X be a Tychonoff space. &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cbeta%20X$%7d%25.png); denotes the Stone-Cech compactification of X. X is called c-realcompact if for every &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$p%5cin%5cbeta%20X%5csetminus%20X$%7d%25.png);, there exists a normal lower semicontinuous function f on &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cbeta%20X$%7d%25.png); such that f(p) = 0 and f is positive on X.

*Property [#k63d2d27]
-A Tychonoff space X is c-realcompact iff for every &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$p%5cin%5cbeta%20X%5csetminus%20X$%7d%25.png);, there is a sequence of [[regular closed>regular open]] subsets of &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cbeta%20X$%7d%25.png); which satisfies &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$p%5cin%20%5ccap_%7bn%5cin%20%5comega%7dA_n$%7d%25.png); and &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5ccap_%7bn%5cin%20%5comega%7d%28A_n%5ccap%20X%29=%5cemptyset$%7d%25.png);.

*Reference [#q8e340fa]
-Nancy Dykes, ''Generalizations of Realcompact Spaces'', Pacific Journal of Mathematics Vol. 33, No. 3, 1970.
-Mary Anne Swardson and Paul J. Szeptycki, ''When X^* is a P' space'', Canad. Math. Bull. Vol. 39 (4), 1996 pp. 476-485.