Abstract
In pointfree topology, the point-finite covers introduced by Dowker and Strauss do not behave similarly to their classical counterparts with respect to tran- sitive quasi-uniformities, contrarily to what happens with other familiar types of interior-preserving covers. The purpose of this paper is to remedy this by modifying the definition of Dowker and Strauss. We present arguments to justify that this modification turns out to be the right pointfree definition of point-finiteness. Along the way we place point-finite covers among the classes of interior-preserving and closure-preserving families of covers that are relevant for the theory of (transitive) quasi-uniformities, completing the study initiated with Ferreira and Picado, Kyungpook Math. J., 44: 415–442, 2004.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banaschewski, B.: The real numbers in pointfree topology. In: Textos de Matemática, Série B, vol. 12. University of Coimbra, Coimbra (1997)
Brümmer, G.C.L.: Functorial transitive quasi-uniformities. In: Categorical Topology, Proceedings Conference, Toledo, 1983, pp. 163–184. Heldermann Verlag, Berlin (1984)
Chen, X.: On the paracompactness of frames. Comment. Math. Univ. Carolin. 33, 485–491 (1992)
Dowker, C.H., Strauss, D.: Paracompact frames and closed maps. In: Symposia Mathematica, Convegno sulla Topologia Insiemistica e Generale, INDAM, Rome, 1973, vol. 16, pp. 93–116. Academic, London (1975)
Engelking, R.: General topology. In: Sigma Series in Pure Math., vol. 6. Heldermann Verlag, Berlin (1989)
Ferreira, M.J., Picado, J.: A method of constructing compatible quasi-uniformities for an arbitrary frame. Kyungpook Math. J. 44, 415–442 (2004)
Ferreira, M.J., Picado, J.: Functorial quasi-uniformities on frames. Appl. Categ. Structures 13, 281–303 (2005)
Fletcher, P.: On totally bounded quasi-uniform spaces. Arch. Math. 21, 396–401 (1970)
Fletcher, P., Lindgren, W.F.: Quasi-uniform spaces. Marcel Dekker, New York (1982)
Isbell, J.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)
Johnstone, P.T.: Stone spaces. In: Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge, UK (1982)
Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. In: Memoirs AMS, vol. 309. AMS, Providence, RI (1984)
Lefschetz, S.: Algebraic topology. In: AMS Colloquium Publ., vol. 27. AMS, New York (1942)
Picado, J., Pultr, A., Tozzi, A.: Locales In: Pedicchio, M.C., Tholen, W. (eds.) Categorical Foundations – Special Topics in Order, Topology, Algebra and Sheaf Theory. Encyclopedia of Mathematics and its Applications, vol. 97, pp. 49–101. Cambridge University Press, Cambridge, UK (2003)
Plewe, T.: Higher order dissolutions and Boolean coreflections of locales. J. Pure Appl. Algebra 154, 273–293 (2000)
Plewe, T.: Sublocale lattices. J. Pure Appl. Algebra 168, 309–326 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Bernhard Banaschewski on the occasion of his 80th birthday.
Rights and permissions
About this article
Cite this article
Ferreira, M.J., Picado, J. On Point-finiteness in Pointfree Topology. Appl Categor Struct 15, 185–198 (2007). https://doi.org/10.1007/s10485-006-9039-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-006-9039-2
Key words
- frame
- locale
- sublocale lattice
- congruence lattice
- cover
- interior-preserving cover
- closure-preserving cover
- point-finite cover
- locally finite cover