Abstract
The universality property plays an important role in the field of frames and the notion of saturated class of frames is combined with this property (see Dube et al. (Topology and its Applications 160, 2454–2464, 2013); Iliadis (Topology and its Applications 179, 99–110, 2015) and Iliadis (Topology and its Applications 201, 92–109, 18)). In this paper, we continue such a study, introducing and studying the notion of saturated class of bases for frames. Based on the notions of the small inductive dimension, frind, for frames, which is inserted in Georgiou et al. (2019), and the saturated class of bases, we define the base dimension like-function of the type frind for frames, and prove that in a class of bases which is characterized by this dimension there exist universal elements.
Similar content being viewed by others
References
Banaschewski, B., Gilmour, G.: Stone-Čech compactification and dimension theory for regular σ-frames. J. London Math. Soc. 39(2), 1–8 (1989)
Banaschewski, B.: Universal zero-dimensional compactifications. Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), World Sci. Publ., Teaneck, NJ, 257–269 (1989)
Berghammer, R., Winter, M.: Order-and graph-theoretic investigation of dimensions of finite topological spaces and Alexandroff spaces. Monatshefte für Mathematik. https://doi.org/10.1007/s00605-019-01261-1 (2019)
Brijlall, D., Baboolal, D.: Some aspects of dimension theory of frames. Indian J. Pure Appl. Math. 39(5), 375–402 (2008)
Brijlall, D., Baboolal, D.: The katětov-morita theorem for the dimension of metric frames. Indian J. Pure Appl. Math. 41(3), 535–553 (2010)
Charalambous, M. G.: Dimension theory of σ-frames. J. London Math. Soc. 8(2), 149–160 (1974)
Dube, T., Iliadis, S., van Mill, J., Naidoo, I.: Universal frames. Topology and its Applications 160, 2454–2464 (2013)
Engelking, R.: Theory of dimensions, finite and infinite. Sigma series in pure mathematics, vol. 10. Heldermann Verlag, Lemgo (1995)
Español, L., Gutiérrez, G. J., Kubiak, T.: Separating families of locale maps and localic embeddings. Algebra Universalis 67, 105–112 (2012)
Georgiou, D., Iliadis, S., Megaritis, A., Sereti, F.: Small inductive dimension and universality on frames. Accepted for publication in Algebra Universalis (2019)
Georgiou, D., Kougias, I., Megaritis, A., Prinos, G., Sereti, F.: A study of a new dimension for frames. Accepted for publication in Topology and its Applications (2019)
Georgiou, D. N., Megaritis, A. C., Sereti, F.: A topological dimension greater than or equal to the classical covering dimension. Houst. J. Math. 43(1), 283–298 (2017)
Gevorgyan, P. S., Iliadis, S. D., Sadovnichy, Y.V.: Universality on frames. Topology and its Applications 220, 173–188 (2017)
Iliadis, S.: A constuction of containing spaces. Topology and its Applications 107, 97–116 (2000)
Iliadis, S. D.: Universal spaces and mappings North-Holland mathematics studies, vol. 198. Elsevier Science B.V., Amsterdam (2005)
Iliadis, S. D.: Universal regular and completely regular frames. Topology and its Applications 179, 99–110 (2015)
Iliadis, S. D.: Dimension and universality on frames. Topology and its Applications 201, 92–109 (2016)
Isbell, J. R.: Graduation and Dimension in Locales. In: Aspects of Topology (in Memory of Hugh Dowker 1912–1982). London Math. Soc. Lecture Note Ser., Vol. 93, pp 195–210. Cambridge Univ. Press, Cambridge (1985)
Menger, K.: Über die Dimensionalität von Punktmengen, Erster. Teil Monatshefte für Mathematik und Physik 33, 148–160 (1923)
Menger, K.: Über die Dimension von Punktmengen, II. Teil. Monatshefte für Mathematik und Physik 34, 137–161 (1926)
Pears, A. R.: Dimension theory of general spaces. Cambridge University Press, Cambridge (1975)
Picardo, J., Pultr, A.: Frames and Locales. Topology without points. Frontiers in mathematics. Birkhäuser/Springer, Basel (2012)
Sancho de Salas, J. B., Sancho de Salas, M. T.: Dimension of distributive lattices and universal spaces. Topology and its Applications 42, 25–36 (1991)
Vinokurov, V.G.: A lattice method of defining dimension. Dokl. Akad. Nauk SSSR 168(3), 663–666 (1966). (Russian)
Acknowledgements
The authors would like to thank the reviewer for the careful reading of the paper and the useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The fourth author of this paper F. Sereti (with Scholarship Code: 2547) would like to thank the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) for the financial support of this study.
Rights and permissions
About this article
Cite this article
Georgiou, D., Iliadis, S., Megaritis, A. et al. Universality Property and Dimension for Frames. Order 37, 427–444 (2020). https://doi.org/10.1007/s11083-019-09513-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-019-09513-3