Notes on the spatial part of a frame

Document Type : Research Paper

Authors

1 School of Computer Science, University of Birmingham, B15 2TT Birmingham, UK

2 Department of Mathematics University of Coimbra PORTUGAL

3 Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské ném. 24, 11800 Praha 1, Czech Republic

10.48308/cgasa.2023.233584.1435

Abstract

A locale (frame) L has a largest spatial sublocale generated by the primes (spectrum points), the spatial part SpL. In this paper we discuss some of the properties of the embeddings SpL ⊆ L. First we analyze the behaviour of the spatial parts in the assembly: the points of L and of S(L)^op (∼=
the congruence frame) are in a natural one-one correspondence while the topologies of SpL and Sp(S(L)^op) differ. Then we concentrate on some special types of embeddings of SpL into L, namely in the questions when SpL is complemented, closed, or open. While in the first part L was general, here we need some restrictions (weak separation axioms) to obtain suitable formulas

Keywords

Main Subjects

References

[1] Arrieta, I., On joins of complemented sublocales, Algebra Universalis 83 (2022), Art. no. 1, 11 pp.
[2] Arrieta, I., “A study of localic subspaces, separation, and variants of normality and their duals”, Ph.D. Thesis, University of Coimbra and University of the Basque Country UPV/EHU, 2022.
[3] Aull, C.E. and Thron, W.J., Separation axioms between T0 and T1, Indag. Math. 24 (1963), 26-37.
[4] Banaschewski, B. and Pultr, A., Variants of openness, Appl. Categ. Structures 2 (1994), 331-350.
[5] Banaschewski, B. and Pultr, A., On covered prime elements and complete homomorphisms of frames, Quaest. Math. 37 (2014), 451-454.
[6] Clementino, M.M., Picado, J., and Pultr, A., The other closure and complete sublocales, Appl. Categ. Structures 26 (2018), 891-906; errata ibid 907-908.
[7] Ern´e, M., Picado, J., and Pultr, A., Adjoint maps between implicative semilattices and continuity of localic maps, Algebra Universalis 83 (2022), Art. no. 13, 23 pp.
[8] Ferreira, M.J., Pinto, S., and Picado, J., Remainders in point-free topology, Topology Appl. 245 (2018), 21-45.
[9] Isbell, J.R., Atomless parts of spaces, Math. Scand. 31 (1972), 5-32.
[10] Isbell, J.R., First steps in descriptive theory of locales, Trans. Amer. Math. Soc. 327 (1991), 353-371; errata ibid 341 (1994), 467-468.
[11] Johnstone, P.T., “Stone Spaces”, Cambridge University Press, 1982.
[12] Johnstone, P.T. and Sun, S.-H., Weak products and Hausdorff locales, in: Categorical Algebra and its applications, Lecture Notes in Math. 1348 (1988), 173-193.
[13] Paseka, J. and ˇSmarda, B., T2-frames and almost compact frames, Czechoslovak Math. J. 42 (1992), 297-313.
[14] Picado, J. and Pultr, A., “Frames and Locales: topology without points”, Frontiers in Mathematics 28, Birkh¨auser-Springer, 2011.
[15] Picado, J. and Pultr, A., “Separation in Point-Free Topology”, Birkh¨auser-Springer, 2021.
[16] Picado, J. and Pultr, A., Notes on sublocales and dissolution, DMUC Preprint 23-26, University of Coimbra, (2023) (submitted).
[17] Picado, J., Pultr, A., and Tozzi, A., Joins of closed sublocales, Houston J. Math. 45 (2019), 21-38.
[18] Plewe, T., Higher order dissolutions and Boolean coreflections of locales, J. Pure Appl. Algebra 154 (2000), 273-293.
[19] Rosick´y, J. and ˇSmarda, B., T1-locales, Math. Proc. Cambridge Phil. Soc. 98 (1958), 81-86.
[20] Simmons, H., A framework for topology, in: “Logic Colloquium ’77”, Studies in Logic and the Foundations of Mathematics 96 (1978), 239–251.
[21] Simmons, H., The lattice theoretic part of topological separation properties, Proc. Edinburgh Math. Soc. 21(2) (1978), 41-48.
[22] Suarez, A.L., On the relation between subspaces and sublocales, J. Pure Appl. Alg. 226 (2022), Art. no. 106851, 24 pp.
[23] Thron, W.J., Lattice-equivalence of topological spaces, Duke Math. J. 29 (1962), 671-679.
Categories and General Algebraic Structures with Applications
Volume 20, Issue 1
Special Issue Dedicated to Prof. Themba Dube
January 2024
Pages 105-129

Files

Share

How to cite

Statistics