Order isomorphisms between bases of topologies
DOI:
https://doi.org/10.17398/2605-5686.37.1.139Keywords:
Lattices, complete metric spaces, locally compact spaces, open regular sets, partially ordered setsAbstract
In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have been able to show some results about arbitrary bases in complete metric spaces and also about regular open subsets in Hausdorff regular topological spaces.
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P.S. Alexandroff, Sur les ensembles de la première classe et les ensembles abstraits, C. R. Acad. Sci. Paris 178 (1924), 185 – 187.
F. Cabello Sánchez, Homomorphisms on lattices of continuous functions, Positivity 12 (2) (2008), 341 – 362.
F. Cabello Sánchez, J. Cabello Sánchez, Some preserver problems on algebras of smooth functions, Ark. Mat. 48 (2) (2010), 289 – 300.
F. Cabello Sánchez, J. Cabello Sánchez, Nonlinear isomorphisms of lattices of Lipschitz functions, Houston J. Math. 37 (1) (2011), 181 – 202.
F. Cabello Sánchez, J. Cabello Sánchez, Lattices of uniformly continuous functions, Topology Appl. 160 (1) (2013), 50 – 55.
J. Cabello Sánchez, A sharp representation of multiplicative isomorphisms of uniformly continuous functions, Topology Appl. 197 (2016), 1 – 9.
J. Cabello Sánchez, J.A. Jaramillo, A functional representation of almost isometries, J. Math. Anal. Appl. 445 (2) (2017), 1243 – 1257.
H.G. Dales, F.K. Dashiell, A.T.-M. Lau, D. Strauss, “ Banach Spaces of Continuous Functions as Dual Spaces ”, Springer, Cham, 2016.
A. Daniilidis, J.A. Jaramillo, F. Venegas, Smooth semi-Lipschitz functions and almost isometries between Finsler manifolds, J. Funct. Anal. 279 (8) (2020), 108662, 29 pp.
M.I. Garrido, J.A. Jaramillo, A Banach-Stone theorem for uniformly continuous functions, Monatsh. Math. 131 (2000), 189 – 192.
M.I. Garrido, J.A. Jaramillo, Variations on the Banach-Stone theorem, Extracta Math. 17 (3) (2002), 351 – 383.
M.I. Garrido, J.A. Jaramillo, Homomorphisms on function lattices, Monatsh. Math. 141 (2) (2004), 127 – 146.
I. Gel’fand, A.M. Kolmogorov, On rings of continuous functions on topological spaces, Dokl. Akad. Nauk. SSSR 22 (1) (1939), 11 – 15.
J. Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. (Basel) 85 (2) (2005), 190 – 196.
F. Hausdorff, Die Mengen Gδ in vollständigen Räumen, Fund. Math., 6 (1) (1924), 146 – 148.
M. Hušek, Lattices of uniformly continuous functions determine their sublattices of bounded functions, Topology Appl. 182 (2015), 71 – 76.
A. Jiménez-Vargas, M. Villegas-Vallecillos, Order isomorphisms of little Lipschitz algebras, Houston J. Math. 34 (2008), 1185 – 1195.
I. Kaplansky, Lattices of continuous functions, Bull. Amer. Math. Soc. 53 (1947), 617 – 623.
I. Kaplansky, Lattices of continuous functions, II, Amer. J. Math. 70 (1948), 626 – 634.
M.M. Lavrentieff, Contribution à la théorie des ensembles homéomorphes, Fund. Math. 6 (1924), 149 – 160.
D.H. Leung, W.-K. Tang, Nonlinear order isomorphisms on function spaces, Dissertationes Math. 517 (2016), 1 – 75.
A.N. Milgram, Multiplicative semigroups of continuous functions, Duke Math. J. 16 (1949), 377 – 383.
J. Mrčun, On isomorphisms of algebras of smooth functions, Proc. Amer. Math. Soc. 133 (10) (2005), 3109 – 3113.
T. Shirota, A generalization of a theorem of I. Kaplansky, Osaka Math. J. 4 (1952), 121 – 132.
H.E. Vaughan, On locally compact metrisable spaces, Bull. Amer. Math. Soc. 43 (8) (1937), 532 – 535.
R.C. Walker, “ The Stone-Čech Compactification ”, Paper 566, Carnegie Mellon University, 1972.
S. Willard, “ General Topology ”. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970.
