×

Strong proximal continuity and convergence. (English) Zbl 1287.54022

In this paper, after revising the various concepts of continuity in uniform and proximal spaces, the authors introduce several forms of strong proximal convergence, and investigate the connections in the setting of Tychonoff spaces with compatible uniformities and proximities by comparing them with uniform convergences and by studying them on bornologies and by making a complete comparison among them as well. The notions of strong proximal continuity and strong proximal convergence, as natural adaptations of the similar ones in the uniform case, together with the notion of bornology play a central role.
Strong proximal continuity relies to various convergences on bornologies, in particular when the bornologies have special features or when considering strong proximally continuous functions. The authors show that the notions of strong proximally continuity (convergence) and strong proximally continuity (convergence) restricted to a bornology are equivalent to continuity (convergence) on hyperspaces with suitable topologies. They connect uniform and proximal continuity of functions with the behaviour of natural functors associated with them in convenient hyperspaces, when carrying the proximal topology in the proximal setting or the Hausdorff-Bourbaki uniformity in the uniform case.

MSC:

54E05 Proximity structures and generalizations
54E15 Uniform structures and generalizations
54C05 Continuous maps
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)

References:

[1] Beer, G.; Levi, S., Strong uniform continuity, Journal of Mathematical Analysis and Applications, 350, 2, 568-589 (2009) · Zbl 1161.54003 · doi:10.1016/j.jmaa.2008.03.058
[2] Caserta, A.; Di Maio, G.; Holá, L., Arzelà’s Theorem and strong uniform convergence on bornologies, Journal of Mathematical Analysis and Applications, 371, 1, 384-392 (2010) · Zbl 1202.54004 · doi:10.1016/j.jmaa.2010.05.042
[3] Bouleau, N., Une structure uniforme sur un espace F(E; F), Cahiers de Topologie et Géométrie Différentielle Catégoriques, 11, 2, 207-214 (1969) · Zbl 0189.23402
[4] Bouleau, N., On the coarsest topology preserving continuity
[5] Beer, G., The Alexandroff property and the preservation of strong uniform continuity, Applied General Topology, 11, 2, 117-133 (2010) · Zbl 1252.54004
[6] Gagrat, M. S.; Naimpally, S. A., Proximity approach to extension problems, Fundamenta Mathematicae, 71, 1, 63-76 (1971) · Zbl 0211.25803
[7] Leader, S., On the completion of proximity spaces by local clusters, Fundamenta Mathematicae, 48, 201-215 (1960) · Zbl 0095.37202
[8] Njastad, O., Some properties of proximity and generalized uniformity, Mathematica Scandinavica, 12, 47-56 (1963) · Zbl 0121.17502
[9] Di Concilio, A.; Naimpally, S. A., Proximal convergence, Monatshefte für Mathematik, 103, 2, 93-102 (1987) · Zbl 0607.54013 · doi:10.1007/BF01630679
[10] Di Concilio, A.; Naimpally, S., Proximal set-open topologies, Bollettino della Unione Matematica Italiana B, 3, 1, 173-191 (2000) · Zbl 0942.54012
[11] Di Maio, G.; Meccariello, E.; Naimpally, S., Hyper-continuous convergence in function spaces, Questions and Answers in General Topology, 22, 2, 157-162 (2004) · Zbl 1066.54019
[12] Di Maio, G.; Meccariello, E.; Naimpally, S., Hyper-continuous convergence in function spaces. II, Ricerche di Matematica, 54, 1, 245-254 (2005) · Zbl 1387.54012
[13] Di Maio, G.; Meccariello, E.; Naimpally, S. A., A natural functor for hyperspaces, Topology Proceedings, 29, 2, 385-410 (2005) · Zbl 1122.54006
[14] Beer, G.; Levi, S., Pseudometrizable bornological convergence is Attouch-Wets convergence, Journal of Convex Analysis, 15, 2, 439-453 (2008) · Zbl 1173.54002
[15] Beer, G.; Costantini, C.; Levi, S., Bornological convergence and shields, Mediterranean Journal of Mathematics, 1-32 (2011) · Zbl 1275.54003 · doi:10.1007/s00009-011-0162-4
[16] Naimpally, S.; Warrack:, B. D., Proximity Spaces (2008), London, UK: Cambridge University Press, London, UK · Zbl 1184.54027
[17] Di Concilio, A.; Mynard, F.; Elliott, E., Proximity: a powerful tool in extension theory, function spaces, hyperspaces, boolean algebras and point-free geometry, Beyond Topology. Beyond Topology, Contemporary Mathematics, 486, 89-112 (2009), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 1192.54010
[18] Naimpally, S., Proximity Approach to Problems in Topology and Analysis (2009), München, Germany: Oldenbourg, München, Germany · Zbl 1185.54001
[19] Naimpally, S. A., Graph topology for function spaces, Transactions of the American Mathematical Society, 123, 267-272 (1966) · Zbl 0151.29703 · doi:10.1090/S0002-9947-1966-0192466-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.