Aspects of RG-spaces
Keywords:Almost Baire spaces, RG-spaces, Blumberg spaces, Almost resolvable spaces, Spaces of countable pseudocharacter, Prime zideal, P-space, Almost-P space
A Tychonoff space X which satisfies the property that G(X) = C(Xδ) is called an RG-space, where G(X) is the minimal regular ring extension of C(X) inside F(X), the ring of all functions from X to R, and Xδ is the topology on X generated by its Gδ-sets. We correct an error tha twe found in the proof of and show that RG-spaces must satisfy a finite dimensional condition.
We also introduce a new class of topological spaces which we call almost k-Baire spaces. The class of almost Baire spaces is a particular instance. We show that every RG-space is an almost Baire space but not necessarily a Baire space. However RG-spaces of countable pseudocharacter must be Baire and, furthermore, their dense sets have dense interiors.
F. Abohalfya, On RG-spaces and the space of prime d-ideals in C(X), Phd Thesis, Concordia University, (2010).
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
R. Bolstein, Sets of points of discontinuity, Proc. Amer. Math. Soc. 38 (1960), 193–197. http://dx.doi.org/10.1090/S0002-9939-1973-0312457-9
J. C. Bradford and C. Goffman, Metric spaces in which Blumberg’s theorem holds, Proc. Amer. Math. Soc. 11 (1960), 667–670.
W. D. Burgess and R. Raphael, The regularity degree and epimorphisms in the category of commutative rings, Comm. Algebra 29, no. 6 (2001), 2489–2500. http://dx.doi.org/10.1081/AGB-100002403
G. H. Butcher, An extension of the sum theorem of dimension theory, Duke Math. 18 (1951), 859–874. http://dx.doi.org/10.1215/S0012-7094-51-01881-9
C. Bandyopadhyay and C. Chattopadhyay, On resolvable and irresolvable spaces, Internat. J. Math. Sci 16, no. 4 (1993), 657–662. http://dx.doi.org/10.1155/S016117129300081X
R. Fox and R. Levy, A Baire space with first category Gδ -topology, Top. Proc. 9, no. 2 (1984), 293–295.
M. Ganster, Pre-open sets and resolvable spaces, Kyungpook Math. J. 27 (1987), 135–143.
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, Comm. Math. Univ. Carol. 44, no. 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. Lambek, Lectures on rings and modules, Blaisdell, Toronto, 1966.
R. Levy, Strongly non-Blumberg spaces, General Topology and Appl. 4 (1974), 173– 177. http://dx.doi.org/10.1016/0016-660X(74)90019-1
R. Levy, Almost P-spaces, Can. J. Math. 29 (1977), 284–288. http://dx.doi.org/10.4153/CJM-1977-030-7
J. Martinez and E. Zenk, Dimension in Algebraic Frames II: Applications of Frames of Ideals in C(X), Comm. Math. Univ. Carol. 46, no. 4 (2005), 607–636.
M. Henriksen, R. Raphael and R. G. Woods, A minimal regular ring extension of C(X), Fund. Math. 172, no. 1 (2002), 1–17. http://dx.doi.org/10.4064/fm172-1-1
R. Raphael and R. G. Woods, The epimorphic hull of C(X), Top. Appl. 105 (2000), 65–88. http://dx.doi.org/10.1016/S0166-8641(99)00036-X
R. Raphael and R. G. Woods, On RG-spaces and the regularity degree, Appl. Gen. Topol. 7 (2006), 72–101.
F. D. Tall, The countable chain condition versus separability-applications of Martin’’s axiom, General Topology Appl. 4 (1974), 315–339. http://dx.doi.org/10.1016/0016-660X(74)90010-5
A. Tamariz-Mascara and H. Villegas-Rodriguez, Spaces of continuous functions, box products, and almost-ω resolvable spaces, Comment. Math. Univ. Carolinae 43, no. 4 (2002), 687–705.
W. A. R. Weiss, A solution to the Blumberg problem, Bull. Amer. Math. Soc. 81 (1975), 957–958. http://dx.doi.org/10.1090/S0002-9904-1975-13914-0
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.