More on Uniform Paracompactness in Pointfree Topology
-
Themba Dube
and
Inderasan Naidoo
Abstract
We revisit uniformly paracompact uniform frames and show that, in analogy with their spatial counterparts, they have a characterisation in terms of a “completeness property”. Namely, they are precisely those in which every weakly Cauchy filter clusters. We also give another characterisation in terms of the Čech-Stone compactification of the underlying frame. By tweaking the definition of uniformly paracompact frames, we define uniformly para-Lindelöf frames (analogously to same-named uniform spaces) and characterise them in terms of the Lindelöf coreflection of the underlying frame. This latter characterisation has no spatial counterpart.
References
[1] BANASCHEWSKI, B.-GILMOUR, C.: Stone-Čech compactification and dimension theory for σ-frames, J. Lond. Math. Soc. (2) 39 (1989), 1-8.10.1112/jlms/s2-39.1.1Search in Google Scholar
[2] BANASCHEWSKI, B.-GILMOUR, C.: Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 577-587.Search in Google Scholar
[3] BANASCHEWSKI, B.-HONG, S. S.: Extension by continuity in pointfree topology, Appl. Categ. Structures 8 (2000), 475-486.10.1023/A:1008673803010Search in Google Scholar
[4] BANACHEWSKI, B.-PULTR, A.: Paracompactness revisited, Appl. Categ. Structures 1 (1993), 181-190.10.1007/BF00880042Search in Google Scholar
[5] DUBE, T.: Notes on uniform frames, Quaest. Math. 27 (2004), 9-20.10.2989/16073600409486079Search in Google Scholar
[6] DUBE, T.: A broader view of the almost Lindel¨of property, Algebra Universalis 65 (2011), 263-276.10.1007/s00012-011-0127-2Search in Google Scholar
[7] ESCARDÓ, M. H.: The regular-locally compact coreflection of a stably locally compact locale, J. Pure Appl. Algebra 157 (2005), 41-55.10.1016/S0022-4049(99)00172-3Search in Google Scholar
[8] FLETCHER, P.-LINDGREN, W. F.: C-complete quasi-uniform spaces, Arch. Math. (Basel) 30 (1978), 175-180.10.1007/BF01226037Search in Google Scholar
[9] FROLÍK, Z.: On paracompact uniform spaces, Czechoslovak Math. J. 33(108) (1983), 476-485.10.21136/CMJ.1983.101897Search in Google Scholar
[10] GANDINI, P. M.: On uniformly para-Lindel¨of spaces, Rend. Istit. Mat. Univ. Trieste 19 (1987), 183-188.Search in Google Scholar
[11] HOHTI, A.: On uniform paracompactness, Ann. Acad. Sci. Fenn. Math. Diss. 36 (1981).Search in Google Scholar
[12] HONG, S. S.: Convergence in Frames, Kyungpook Math. J. 35 (1995), 85-91.Search in Google Scholar
[13] HOWES, N. R.: On completeness, Pacific J. Math. 38 (1971), 431-440.10.2140/pjm.1971.38.431Search in Google Scholar
[14] HOWES, N. R.: Modern Analysis and Topology Universitext, Springer, London, 1995.10.1007/978-1-4612-0833-4Search in Google Scholar
[15] MADDEN, J.-VERMEER, H.: Lindel¨of locales and realcompactness, Math. Proc. Cambridge Philos. Soc. 99 (1986), 473-480.10.1017/S0305004100064410Search in Google Scholar
[16] MARCONI, U.: On the uniform paracompactness, Rend. Semin. Mat. Univ. Padova 72 (1984), 319-328.Search in Google Scholar
[17] NAIDOO, I.: Strong Cauchy completeness in uniform frames, Acta Math. Hungar. 116 (2007), 273-284.10.1007/s10474-007-6053-2Search in Google Scholar
[18] NAIDOO, I.: On preparacompactness of uniform frames, Acta Math. Hungar. 132 (2011), 253-268.10.1007/s10474-011-0094-2Search in Google Scholar
[19] NAIDOO, I.: On completion in the category SSNσFrm, Math. Slovaca 63 (2013), 201-214.10.2478/s12175-012-0093-ySearch in Google Scholar
[20] PICADO, J.-PULTR, A.: Frames and Locales: topology without points. Front. Math., Springer, Basel, 2012.10.1007/978-3-0348-0154-6Search in Google Scholar
[21] RICE, M. D.: A note on uniform paracompactness, Proc. Amer. Math. Soc. 62 (1977), 359-362.10.1090/S0002-9939-1977-0436085-3Search in Google Scholar
[22] VERMEULEN, J. J. C.: Proper maps of locales, J. Pure Appl. Algebra 92 (1994), 79-107.10.1016/0022-4049(94)90047-7Search in Google Scholar
[23] WEI HE-MAOKANG LUO: Completely regular proper reflection of locales over a given locale, Proc. Amer. Math. Soc. 141 (2013), 403-408. 10.1090/S0002-9939-2012-11329-2Search in Google Scholar
© Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Complexities of Relational Structures
- Notes on the Product of Locales
- Free Objects and Free Extensions in the Category of Frames
- More on Uniform Paracompactness in Pointfree Topology
- Nonmeasurable Cardinals and Pointfree Topology
- Pseudocompact σ-Frames
- Torsion Radicals and Torsion Classes of Cyclically Ordered Groups
- Generalized Commutativity of Lattice-Ordered Groups
- The Relatively Uniform Completion, Epimorphisms and Units, in Divisible Archimedean L-Groups
- Polymorphism-Homogeneous Monounary Algebras
- αcc-Baer Rings
- Pushout Invariance Revisited
- Testing Statistical Hypotheses in Singular Weakly Nonlinear Regression Models
Articles in the same Issue
- Complexities of Relational Structures
- Notes on the Product of Locales
- Free Objects and Free Extensions in the Category of Frames
- More on Uniform Paracompactness in Pointfree Topology
- Nonmeasurable Cardinals and Pointfree Topology
- Pseudocompact σ-Frames
- Torsion Radicals and Torsion Classes of Cyclically Ordered Groups
- Generalized Commutativity of Lattice-Ordered Groups
- The Relatively Uniform Completion, Epimorphisms and Units, in Divisible Archimedean L-Groups
- Polymorphism-Homogeneous Monounary Algebras
- αcc-Baer Rings
- Pushout Invariance Revisited
- Testing Statistical Hypotheses in Singular Weakly Nonlinear Regression Models
