Τ-quasi-Cauchy spaces - a non-symmetric theory of completeness and completion

Gunther Jäger

doi:10.4995/agt.2023.18783

Authors

Gunther Jäger University of Applied Sciences Stralsund, Germany https://orcid.org/0000-0002-1495-4564

DOI:

https://doi.org/10.4995/agt.2023.18783

Keywords:

fuzzy topology, pair Τ-filter, Cuachy pair Τ-filter, Τ-quasi-Cauchy space, Τ-quasi-uniform space, L-metric space

Abstract

Based on the concept of Cauchy pair Τ-filters, we develop an axiomatic theory of completeness for non-symmetric spaces, such as Τ-quasi-uniform (limit) spaces or L-metric spaces. We show that the category of Τ-quasi-Cauchy spaces is topological and Cartesian closed and we construct a finest completion for a non-complete Τ-quasi-Cauchy space. In the special case of symmetry, Τ-quasi-Cauchy spaces can be identified with Τ-Cauchy spaces.

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References

J. J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.

R. Bĕlohlávek, Fuzzy Relation Systems, Foundation and Principles, Kluwer Academic/Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, 2002.

M. M. Clementino and D. Hofmann, Lawvere completeness in topology, Appl. Categ. Struct. 17 (2009), 175-210. https://doi.org/10.1007/s10485-008-9152-5

D. Doitchinov, Completeness in quasi-metric spaces, Topology Appl. 30 (1988), 127-148. https://doi.org/10.1016/0166-8641(88)90012-0

R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37 (1997), 257-276. https://doi.org/10.1007/s000120050018

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

J. Gutiérrez-García, On stratified L-valued filters induced by Τ-filters, Fuzzy Sets and Systems 157 (2006), 813-819. https://doi.org/10.1016/j.fss.2005.09.003

J. He, H. Lai and L. Shen, Towards probabilistic partial metric spaces: Diagonals between distance distributions, Fuzzy Sets and Systems 370 (2019), 99-119. https://doi.org/10.1016/j.fss.2018.07.011

U. Höhle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math. 38 (1982), 289-323. https://doi.org/10.1007/BF01170928

U. Höhle, Commutative, residuated l-monoids, in: Non-classical logics and their applications to fuzzy subsets, U. Höhle, E. P. Klement, eds., Kluwer, Dordrecht (1995), 53-106. https://doi.org/10.1007/978-94-011-0215-5_5

D. Hofmann and W. Tholen, Lawvere Completion and separation via closure, Appl. Categ. Struct. 18 (2010), 259-287. https://doi.org/10.1007/s10485-008-9169-9

D. Hofmann, G. J. Seal and W. Tholen, Monoidal topology, Cambridge University Press 2014. https://doi.org/10.1017/CBO9781107517288

G. Jäger and Y. Yue, Τ-uniform convergence spaces, Iranian J. Fuzzy Syst. 19, no. 2 (2022), 133-149.

G. Jäger, Diagonal conditions and uniformly continuous extension in Τ-uniform limit spaces, Iranian J. Fuzzy Syst. 19, no. 5 (2022), 131-145.

G. Jäger, Sequential completeness for Τ-quasi-uniform spaces and a fixed point theorem, Mathematics 10 (2022): 2285. https://doi.org/10.3390/math10132285

H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Annal. 176 (1968), 334-341. https://doi.org/10.1007/BF02052894

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

F. W. Lawvere, Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973) 135-166. Reprinted in: Reprints in Theory and Applications of Categories 1 (2002), 1-37. https://doi.org/10.1007/BF02924844

G. Preuβ, Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions, Topology Appl. 156 (2009), 2005 - 2012. https://doi.org/10.1016/j.topol.2009.03.026

E. E. Reed, Completion of uniform convergence spaces, Math. Annal. 194 (1971), 83-108. https://doi.org/10.1007/BF01362537

L. Reid and G. Richardson, Lattice-valued spaces: Τ-completions, Fuzzy Sets and Systems 369 (2019), 1-19. https://doi.org/10.1016/j.fss.2018.06.003

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983.

Y. Wang and Y. Yue, Cauchy completion of fuzzy quasi-uniform Spaces, Filomat 35, no. 12 (2021), 3983-4004. https://doi.org/10.2298/FIL2112983W

Q. Yu and J. Fang, The category of Τ-convergence spaces and its cartesian-closedness, Iranian J. of Fuzzy Systems 14, no. 3 (2017), 121-138.

Y. Yue and J. Fang, Completeness in probabilistic quasi-uniform spaces, Fuzzy Sets and Systems 370 (2019), 34-62. https://doi.org/10.1016/j.fss.2018.08.005

D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems 158 (2007), 349-366. https://doi.org/10.1016/j.fss.2006.10.001