Non-symmetric convergence and regularity
Submitted: 2024-09-16
|Accepted: 2024-11-12
|Published: 2025-04-01
Copyright (c) 2025 Gunther Jäger
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Downloads
Keywords:
Quasi-convergence space, regularity, extension of mappings, continuous convergence, biconvergence, convergence space
Supporting agencies:
Abstract:
We study regularity in quasi-convergence spaces and biconvergence spaces. We show that a notion weaker than the usually considered pairwise regularity is sufficient in important applications. This regularity can be defined in terms of closures of pair filters or by a diagonal condition. We show its appropriateness by characterizing it in terms of continuous convergence and in terms of extensions of continuous mappings.
References:
M. M. Clementino and D. Hofmann, Lawdvere Completeness in Topology, Appl. Categor. Struct. 17 (2009), 175-210. https://doi.org/10.1007/s10485-008-9152-5
E. Colebunders, F. Mynard, and W. Trott, Function Spaces and Contractive Extensions in Approach Theory: The Role of Regularity, Appl. Categor. Struct. 22 (2014), 551-563. https://doi.org/10.1007/s10485-013-9321-z
C. H. Cook, On continuous extensions, Math. Annalen 176 (1968), 302-304. https://doi.org/10.1007/BF02052890
H. R. Fischer, Limesräume, Math. Annalen 137 (1959), 269-303. https://doi.org/10.1007/BF01360965
P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Marcel Dekker, New York 1982.
G. Jäger, T-quasi-Cauchy spaces - a non-symmetric theory of completeness and completion, Appl. Gen. Topol. 24, no. 1 (2023), 205-227. https://doi.org/10.4995/agt.2023.18783
G. Jäger, Quasi-convergence spaces and biconvergence spaces, Topology Proceedings 64 (2024), 83-102.
J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. S3-13 (1963), 71-89. https://doi.org/10.1112/plms/s3-13.1.71
D. C. Kent, Convergence functions and their related topologies, Fund. Math. 54 (1964), 125-133. https://doi.org/10.4064/fm-54-2-125-133
D. C. Kent and G. D. Richardson, Convergence spaces and diagonal conditions, Top. Appl. 70 (1996), 167-174. https://doi.org/10.1016/0166-8641(95)00094-1
H.-J. Kowalsky, Limesräume und Komplettierung, Math. Nachrichten 12 (1954), 301-340. https://doi.org/10.1002/mana.19540120504
H. P. A. Künzi, An introduction to quasi-uniform spaces, in: Beyond Topology (F. Mynard, E. Pearl, eds.), Contempory Mathematics 486, Amer. Math. Soc., Providence, Rhodes Island 2009, 239-304. https://doi.org/10.1090/conm/486/09511
S. Lai, Pairwise concepts in bitopological spaces, J. Austral. Math. Soc. (Ser. A) 26 (1978), 241-250. https://doi.org/10.1017/S1446788700011733
E. P. Lane, Bitopological spaces and quasi-uniform spaces, Proc. London Math. Soc. S3-17 (1967), 241-256. https://doi.org/10.1112/plms/s3-17.2.241
W. F. Lindgren and P. Fletcher, A construction of the pair completion of a quasi-uniform space, Can. Math. Bull. 21 (1978), 53-59. https://doi.org/10.4153/CMB-1978-009-2
H. Poppe, Compactness in general function spaces, VEB Deutscher Verlag der Wissenschaften, Berlin 1974.
W. J. Pervin, Quasi-uniformization of topological spaces, Math. Annalen 147 (1962), 316-317. https://doi.org/10.1007/BF01440953
W. J. Pervin and H. J. Biesterfeld Jr., Uniformization of convergence spaces Part II: Conjugate convergence structures and bi-structures, Math. Annalen 177 (1968), 43-48. https://doi.org/10.1007/BF01350728
G. Preuß, Seminuniform convergence spaces, Math. Japonica 41, no. 3 (1995), 465-491.
