Document Zbl 1530.54007

\(\top\)-quasi-Cauchy spaces – a non-symmetric theory of completeness and completion. (English) Zbl 1530.54007

Appl. Gen. Topol. 24, No. 1, 205-227 (2023).

The author develops an axiomatic theory of completeness for non-symmetric spaces, shows that the category of \(\top\)-quasi-Cauchy spaces is topological and Cartesian closed and constructs a finest completion for a non-complete \(\top\)-quasi-Cauchy space.

Reviewer: Javier Gutiérrez García (Bilbao)

Cited in 1 Document

MSC:

54A40	Fuzzy topology
54A05	Topological spaces and generalizations (closure spaces, etc.)
54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54E15	Uniform structures and generalizations

Keywords:

fuzzy topology; pair \(\top\)-filter; Cauchy pair \(\top\)-filter; \(\top\)-quasi-Cauchy space; \(\top\)-quasi-uniform space; \(\top\)-quasi-uniform limit space; \(\mathsf{L}\)-metric space

Cite Review PDF

Full Text: DOI

OA License

References:

[1]	J. J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.
[2]	R. Bĕlohlávek, Fuzzy Relation Systems, Foundation and Principles, Kluwer Academic/Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, 2002. · Zbl 1067.03059
[3]	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 · Zbl 1173.18001 · doi:10.1007/s10485-008-9152-5
[4]	D. Doitchinov, Completeness in quasi-metric spaces, Topology Appl. 30 (1988), 127-148. https://doi.org/10.1016/0166-8641(88)90012-0 · Zbl 0668.54019 · doi:10.1016/0166-8641(88)90012-0
[5]	R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37 (1997), 257-276. https://doi.org/10.1007/s000120050018 · Zbl 0906.54024 · doi:10.1007/s000120050018
[6]	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 · Zbl 0393.54019 · doi:10.4153/CMB-1978-009-2
[7]	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 · Zbl 1101.54006 · doi:10.1016/j.fss.2005.09.003
[8]	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 · Zbl 1423.54020 · doi:10.1016/j.fss.2018.07.011
[9]	U. Höhle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math. 38 (1982), 289-323. https://doi.org/10.1007/BF01170928 · Zbl 1004.54500 · doi:10.1007/BF01170928
[10]	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 · doi:10.1007/978-94-011-0215-5_5
[11]	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 · Zbl 1207.18006 · doi:10.1007/s10485-008-9169-9
[12]	D. Hofmann, G. J. Seal and W. Tholen, Monoidal topology, Cambridge University Press 2014. https://doi.org/10.1017/CBO9781107517288 · Zbl 1297.18001 · doi:10.1017/CBO9781107517288
[13]	G. Jäger and Y. Yue, Τ-uniform convergence spaces, Iranian J. Fuzzy Syst. 19, no. 2 (2022), 133-149. · Zbl 1505.54012
[14]	G. Jäger, Diagonal conditions and uniformly continuous extension in Τ-uniform limit spaces, Iranian J. Fuzzy Syst. 19, no. 5 (2022), 131-145. · Zbl 1522.54012
[15]	G. Jäger, Sequential completeness for Τ-quasi-uniform spaces and a fixed point theorem, Mathematics 10 (2022): 2285. https://doi.org/10.3390/math10132285 · doi:10.3390/math10132285
[16]	H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Annal. 176 (1968), 334-341. https://doi.org/10.1007/BF02052894 · Zbl 0155.50302 · doi:10.1007/BF02052894
[17]	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 · Zbl 1193.54014 · doi:10.1090/conm/486/09511
[18]	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 · Zbl 0335.18006 · doi:10.1007/BF02924844
[19]	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 · Zbl 1172.54017 · doi:10.1016/j.topol.2009.03.026
[20]	E. E. Reed, Completion of uniform convergence spaces, Math. Annal. 194 (1971), 83-108. https://doi.org/10.1007/BF01362537 · Zbl 0213.49601 · doi:10.1007/BF01362537
[21]	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 · Zbl 1423.54011 · doi:10.1016/j.fss.2018.06.003
[22]	B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983. · Zbl 0546.60010
[23]	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 · doi:10.2298/FIL2112983W
[24]	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. · Zbl 1398.54007
[25]	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 · Zbl 1423.54028 · doi:10.1016/j.fss.2018.08.005
[26]	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 · Zbl 1112.54005 · doi:10.1016/j.fss.2006.10.001

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.