Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-09T07:02:08.114Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Lattice of Primitive Convergence Structures

Published online by Cambridge University Press:  20 November 2018

C. R. Atherton Jr.
C. R. Atherton Jr.*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S be any set and denote by F(S) the collection of all fiters on S. The collection A(S) of all mappings from F(S) to 2s, 2s being ordered by the dual of its usual ordering, may be regarded as a product of complete Boolean algebras and is, therefore, a complete atomic Boolean algebra [4]. A(S) is called the lattice of primitive convergence structures on S. If q ∈ A(S) and , then is said to q-converge to a point x ∈ S if . The collection of all topologies on S may be identified with a subset of A(S); this subset of A(S) will be denoted by T(S). A more specialized class of primitive convergence structures, and one which properly contains T(S), is C(S), the subcomplete lattice of all convergence structures on S.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 392 - 397
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Atherton, C. R., Concerning intrinsic topologies on Boolean algebras and certain bicompactly generated lattices, Glasgow Math. J. 11 (1970), 156–161.Google Scholar
2. Biesterfeldt, H. J., Jr., Completion of a class of uniform convergence spaces, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 602–604.Google Scholar
3. Biesterfeldt, H. J., Jr., Regular convergence spaces, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 605–607.Google Scholar
4. Birkhoff, G., Lattice theory, Third edition, Amer. Math. Soc. Colloq. Publ., Vol. 25 (Amer. Math. Soc, Providence, R. L, 1967).Google Scholar
5. Choquet, G., Convergences, Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N. S.) 28 (1948), 57–112.Google Scholar
6. Cook, C. H. and Fischer, H. R., Regular convergence spaces, Math. Ann. 174 (1967), 1–7.Google Scholar
7. Fischer, H. R., Limesraume, Math. Ann. 137 (1959), 269–303.Google Scholar
8. Frink, O., Ideals in partially ordered sets, Amer. Math. Monthly 61 (1954), 223–233.Google Scholar
9. Hearsey, B., On extending regularity and other topological properties to convergence structures, Unpublished Ph.D. dissertation, Washington State University, Pullman, Washington, 1968.Google Scholar
10. Kent, D. C., Convergence functions and their related topologies, Fund. Math. 54 (1964), 125–133.Google Scholar