On RG-spaces and the regularity degree

R. Raphael, R.G. Woods


We continue the study of a lattice-ordered ring G(X), associated with the ring C(X). Following, X is called RG when G(X) = C(Xδ). An RG-space must have a dense set of very weak P-points. It must have a dense set of almost-P-points if Xδ is Lindelöf, or if the continuum hypothesis holds and C(X) has small cardinality. Spaces which are RG must have finite Krull dimension when taken with respect to the prime z-ideals of C(X). There is a notion of regularity degree defined via the functions in G(X). Pseudocompact spaces and metric spaces of finite regularity degree are characterized.


P-space; Almost-P space; Prime z-ideal; RG-space; Very weak P-point

Full Text:



M. Barr, W. D. Burgess and R. Raphael, Ring epimorphisms and C(X), Theory Appl. Categ. 11(12) (2003), 283–308.

M. Bell, J. Ginsburg and R. G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79 (1978), 37–43. http://dx.doi.org/10.2140/pjm.1978.79.37

R. L. Blair and A. W. Hager, Extensions of zero-sets and real-valued functions, Math. Zeit. 136 (1974), 41–52. http://dx.doi.org/10.1007/BF01189255

W. D. Burgess and R. Raphael, The regularity degree and epimorphisms in the category of commutative rings, Commun. Algebra, 29(6) (2001), 2489–2500. http://dx.doi.org/10.1081/AGB-100002403

W. W. Comfort and A. W. Hager, Estimates for the number of real-valued continuous functions, Trans. Amer. Math. Soc. 150 (1970), 619–631. http://dx.doi.org/10.1090/S0002-9947-1970-0263016-X

W. W. Comfort and S. Negrepontis, Homeomorphs of three subspaces of βN −N, Math. Z. 107 (1968), 53–58. http://dx.doi.org/10.1007/BF01111048

E. van Douwen and H. Zhou, The number of cozero-sets is an ω-power, Topology Appl. 33 (1989), 115–126. http://dx.doi.org/10.1016/S0166-8641(89)80001-X

L. Gillman and M. Jerison, Rings of Continuous functions, (Van Nostrand, Princeton, 1960). http://dx.doi.org/10.1007/978-1-4615-7819-2

M. Henriksen, J. Martinez and R. G. Woods, Spaces X in which all prime z-ideals of C(X) are maximal or minimal, Commentat. Math. Univ. Carol. 44(2) (2003), 261–294.

M. Henriksen, R. Raphael and R. G. Woods, A minimal regular ring extension of C(X), Fund. Math. 172 (2002), 1–17. http://dx.doi.org/10.4064/fm172-1-1

J. Kennison, Structure and costructure for strongly regular rings, J. Pure Appl. Algebra, 5 (1974), 321–332. http://dx.doi.org/10.1016/0022-4049(74)90041-3

K. Kunen, Some points in βN, Math. Proc. Camb. Philos. Soc. 80 (1976), 385–398. http://dx.doi.org/10.1017/S0305004100053032

J. Lambek, Lectures on rings and modules, (Blaisdell, Toronto, 1966).

D. Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1968), 6–127.

R. Levy and M. D. Rice, Normal P-spaces and the Gδ-topology, Colloq. Math. 44 (1981), 227–240.

R. Montgomery, Structures determined by prime ideals of rings of functions, Trans. Amer. Math. Soc. 147 (1970), 367–380. http://dx.doi.org/10.1090/S0002-9947-1970-0256174-4

S. Mrowka, Some set-theoretic constructions in topology, Fund. Math. 94(2) (1977), 83–92.

J.-P. Olivier, Anneaux absolument plats universels et epimorphismes a but reduits, Seminaire Samuel, (1967)–68, 6-01–6-12.

J. R. Porter and R. G. Woods, Extensions and absolutes of Hausdorff spaces, (Springer Verlag, 1988). http://dx.doi.org/10.1007/978-1-4612-3712-9

R. Raphael, Some epimorphic regular contexts, Theory Appl. Categ. 6 (1999), 94–104.

R. Raphael and R. G. Woods, The epimorphic hull of C(X), Topology Appl. 105 (2000), 65–88. http://dx.doi.org/10.1016/S0166-8641(99)00036-X

J. Terasawa, Spaces N [ R and their dimensions, Topology Appl. 11 (1980), 93–102. http://dx.doi.org/10.1016/0166-8641(80)90020-6

T. Terada, On remote points in X − X, Proc. Amer. Math. Soc. 77 (1979), 264–266.

J. Van Mill, Weak P-points in Cech-Stone compactifications, Trans. Amer. Math. Soc. 283 (1982), 657–678.

R. Wiegand, Modules over universal regular rings, Pac. J. Math. 39, (1971), 807–819. http://dx.doi.org/10.2140/pjm.1971.39.807

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM


Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. P-spaces and an unconditional closed graph theorem
Marek Wójtowicz, Waldemar Sieg
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas  vol: 104  issue: 1  first page: 13  year: 2010  
doi: 10.5052/RACSAM.2010.03

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt