Another closure operator on preneighbourhood spaces

Document Type : Research Paper

Author

Department of Mathematical Sciences, University of South Africa, Unisa Science Campus, corner of Christiaan de Wet & Pioneer Avenue, Florida 1709, Johannesburg, Gauteng, South Africa.\\ National Institute for Theoretical and Computational Sciences (NITheCS), South Africa.

10.48308/cgasa.2024.235195.1478

Abstract

The notions of dense, proper, separated or perfect morphisms and hence of compact, Hausdorff or compact Hausdorff are all consequent to good properties of a family of closed morphisms is well known in literature. Deeper consequences like the Tychonoff product theorem or the Stone Čech compactifications follow from richer properties of the set of closed morphisms. The purpose of this paper is to provide a closure operation on a preneighbourhood space so that the resulting set of closed morphisms possess all the properties mentioned above. 

Keywords

Main Subjects


[1] Borceux, F., “Handbook of Categorical Algebra. 1: Basic Category Theory”, Encyclopedia of Mathematics and its Applications, Vol. 50, Cambridge University Press, 1994.
[2] Bourn, D., 3 × 3 lemma and protomodularity, J. Algebra 236(2) (2001), 778-795.
[3] Clementino, M.M., Giuli, E., and Tholen, W., A functional approach to general topology, Categorical foundations, Encyclopedia Math. Appl., Cambridge Univ. Press,Cambridge 97 (2004), 103-163.
[4] Dikranjan, D. and Tholen, W., “Categorical Structure of Closure Operators, With Applications to Topology, Algebra and Discrete Mathematics”, Mathematics and its Applications, Vol. 346, Kluwer Academic Publishers Group, 1995.
[5] Ghosh, P.P., Internal neighbourhood structures, Algebra Universalis 81(2) (2020), Paper No. 12, 53 pages.
[6] Ghosh, P.P., Internal neighbourhood structures II: Closure and closed morphisms, Categ. General Alg. Structures Appl. 18(1) (2023), 155-223.
[7] Giuli, E. and ˇSlapal, J., Neighborhoods with respect to a categorical closure operator, Acta Math. Hungar. 124(1-2) (2009), 1-14.
[8] Goswami, A. and Janelidze, Z., On the structure of zero morphisms in a quasipointed category, Appl. Categ. Structures 25(6) (2017), 1037-1043.
[9] Hofmann, D. and Tholen, W., Lax algebra meets topology, Topology Appl. 159(9) (2012), 2434-2452.
[10] Holgate, D. and ˇSlapal, J., Categorical neighborhood operators, Topology Appl. 158(17) (2011), 2356-2365.
[11] Holgate, D. and ˇSlapal, J., Closure, interior and neighbourhood in a category, Hacet. J. Math. Stat. 47(6) (2018), 1512-1520.
[12] Mac Lane, S., “Categories for the Working Mathematician”, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, 1998.
[13] Monk, J.D., “Introduction to Set Theory”, McGraw-Hill Book Co., 1969.
[14] Picado, J. and Pultr, A., “Frames and Locales: Topology without Points”, Frontiers in Mathematics, Birkh¨auser/Springer Basel AG, 2012.
[15] Razafindrakoto, A., Neighbourhood operators on categories, Ph.D. Thesis, University of Stellenbosch (2012).
[16] Razafindrakoto, A., On coarse and fine neighbourhood operators, Topology Appl. 159(13) (2012), 3067-3079.
[17] Razafindrakoto, A., Neighbourhood operators: additivity, idempotency and convergence, Appl. Categ. Structures 27(6) (2019), 703-721.
[18] Razafindrakoto, A. and Holgate, D., Interior and neighbourhood, Topology Appl. 168 (2014), 144-152.
[19] Razafindrakoto, A. and Holgate, D., A lax approach to neighbourhood operators, Appl. Categ. Structures 25(3) (2017), 431-445.
[20] Richmond, T. and ˇSlapal, J., Neighborhood spaces and convergence, Topology Proc. 35 (2010), 165-175.
[21] ˇSlapal, J., Neighborhoods and convergence with respect to a closure operator, Math. Slovaca 61(5) (2011), 717-732.
[22] ˇSlapal, J., Compactness and convergence with respect to a neighborhood operator, Collect. Math. 63(2) (2012), 123-137.
[23] Vermeulen, J.J.C., Proper maps of locales, J. Pure Appl. Algebra 92(1) (1994), 79-107.
[24] Vermeulen, J.J.C., A note on stably closed maps of locales, J. Pure Appl. Algebra 157 (2001), 335-339.