⊤-Nets and ⊤-Filters

Subject Areas : Transactions on Fuzzy Sets and Systems

Gunther Jager ¹

1 - School of Mechanical Engineering, University of Applied Sciences Stralsund, Stralsund, Germany.

Received: 2022-02-19 Accepted : 2022-02-20 Published : 2022-05-07

Keywords: Fuzzy topology, Convergence, ⊤-filter, ⊤-net, Strong L-topological space.,

Abstract :

In this paper, we develop a theory of ⊤-nets and study their relation to ⊤-filters. We show that convergence in strong L-topological spaces can be described by both ⊤-nets and ⊤-filters and both concepts are equivalent in the sense that definitions and proofs that are given using ⊤-filters can also be given using ⊤-nets and vice versa.

References:

1] R. G. Bartle, Nets and filters in topology, Amer. Math. Monthly, 52 (1955), 551-557.
[2] R. Bělohlávek, Fuzzy Relation Systems, Foundation and Principles, Kluwer Academic/Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, (2002).
[3] N. Bourbaki, General topology, Springer Verlag, Berlin- Heidelberg- New York- London- Paris- Tokyo, (1990).
[4] H. Cartan, Theéorie des filtres, C. R. Acad. Sci. Paris, 205 (1937), 595-598.
[5] J. Fang, Stratified L-ordered convergence structures, Fuzzy Sets and Systems, 161 (2010), 2130-2149.

[6] J. Fang, Relationships between L-ordered convergencestructures and strong L-topologies, Fuzzy Sets and Systems, 161 (2010), 2923-2944.
[7] J. Fang and Y. Yue, ⊤-diagonal conditions and continuous extension theorem, Fuzzy Sets and Systems, 321 (2017), 73-89.
[8] R. C. Flagg, Quantales and continuity spaces, Algebra Univers., 37 (1997), 257-276.
[9] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, Continuous lattices and domains, Fuzzy Sets Cambridge University Press, (2003).
[10] U. Höhle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math., 38 (1982), 289-323.
[11] U. Höhle, Commutative, residuated l-monoids, in: Non-classical logics and their applications to fuzzy subsets (U. Höhle, E. P. Klement, eds.), Kluwer Academic Publishers, Dordrecht, (1995).
[12] U. Höhle and A. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S. E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Kluwer Academic Publishers, Dordrecht, (1999).
[13] D. Hofmann, G. J. Seal and W. Tholen, Monoidal topology, Cambridge University Press, (2014).
[14] G. Jäger, A category of L-fuzzy convergence spaces, Quaest. Math., 24 (2001), 501-517.
[15] G. Jäger, Stratified LMN-convergence tower spaces, Fuzzy Sets and Systems, 282 (2016), 62-73.
[16] G. Jäger and Y. Yue, T-uniform convergence spaces, Iranian J. Fuzzy Systems, to appear. Doi 10.22111/ijfs.2021.6528.
[17] J.  L. Kelley, General topology, Van Nostrand, Princeton, New Jersey, Toronto, London, New York, (1955).
[18] H.  J. Kowalsky, Topological spaces, Acad. Press, New York, (1965).
[19] H. Lai and D. Zhang, Fuzzy topological spaces with conical neighborhood systems, Fuzzy Sets and Systems, 330 (2018), 87-104.
[20] L. Li and Q. Jin, On stratified L-convergence spaces: Pretopological axioms and diagonal axioms, Fuzzy Sets and Systems, 204 (2012), 40-52.
[21] L. Li, Q. Jin and K. Hu, Lattice-valued convergence associated with CNS spaces, Fuzzy Sets and Systems, 370 (2019), 91-98.
[22] R. Lowen, Convergence in fuzzy topological spaces, Gen. Top. Appl., 10 (1979), 147-160.
[23] R. Lowen, The relation between filter and net convergence in fuzzy topological spaces, Fuzzy Mathematics, 4 (1983), 41-52.
[24] E. H. Moore and H. L. Smith, A general theory of limits, Amer. J. Math., 44 (1922), 102-121.
[25] B. M. Pu and Y. M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599.
[26] G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc., 4 (1953), 518-512.

[27] L. Reid, G. Richardson, Connecting ⊤ and lattice-valued convergences, Iran. J. of Fuzzy Syst., 15(4), (2018), 151-169.
[28] W. Yao, Moore-Smith convergence in (LM)-fuzzy topology, Fuzzy Sets and Systems, 190 (2012), 47-62.
[29] Q. Yu, J. Fang, The category of ⊤-convergence spaces and its cartesian-closedness, Iranian J. of Fuzzy Systems, 14(3) (2017), 121-138.
[30] Y. Yue and J. Fang, Completeness in probabilistic quasi-uniform spaces, Fuzzy Sets and Systems, 370 (2019), 34-62.
[31] Y. Yue and J. Fang, The ⊤-filter monad and its applications, Fuzzy Sets and Systems, 382 (2020), 79-97.
[32] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158 (2007), 349-366.