Completeness of fuzzy quasi-pseudometric spaces
Completeness of fuzzy quasi-pseudometric spaces
The purpose of this paper is to present the relations among the completeness of sequences, of filters and of nets in the framework of fuzzy quasi-pseudometric spaces. In particular, we show that right completeness of filters and of sequences are equivalent under special conditions of fuzzy quasi-pseudometrics. By introducing a kind of more general right K-Cauchy nets in fuzzy quasi-pseudometric spaces, the equivalence between the completeness of the nets and the sequential completeness is established.
___
- [1] E. Alemany and S. Romaguera, On right K-sequentially complete quasi-metric spaces,
Acta Math. Hungar. 75, 267–278, 1997.
- [2] N.F. Al-Mayahi and I.H. Radhi, The closure in fuzzy metric (normed) space, Tikrit
J. Pure Sci. 18, 411–412, 2013.
- [3] F. Castro-Company, S. Romaguera and P. Tirado, The bicompletion of fuzzy quasi-
metric spaces, Fuzzy Sets Syst. 166, 56–64, 2011.
- [4] Ş. Cobzaş, Completeness in quasi-pseudometric spaces–a survey, Math. 8, 1279, 2020.
- [5] A.S. Davis, Indexed systems of neighbourhoods for general topological spaces, Amer.
Math. Mon. 68, 886–893, 1961.
- [6] D. Doitchinov, On completeness in quasi-metric spaces, Topology Appl. 30, 127–148,
1988.
- [7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst.
64, 395–399, 1994.
- [8] A. George and P. Veeramani, Some theorems in fuzzy metric spaces, J. Fuzzy Math.
3, 933–940, 1995.
- [9] V. Gregori and J. Ferrer, A note on R. Stoltenberg’s completion theory of quasi-
uniform spaces. Proc. Lond. Math. Soc. 49, 36–36, 1984.
- [10] V. Gregori, A. López-Crevillén, S. Morillas and A. Sapena, On convergence in fuzzy
metric spaces, Topology Appl. 156, 3002–3006, 2009.
- [11] V. Gregori, J.J. Miñana, S. Morillas and A. Sapena, Characterizing a class of com-
pletable fuzzy metric spaces, Topology Appl. 203, 3–11, 2016.
- [12] V. Gregori, S. Morillas and A. Sapena, On completion of fuzzy quasi-metric spaces,
Topology Appl. 153, 886–899, 2005.
- [13] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets
Syst. 115, 485–489, 2000.
- [14] V. Gregori and S. Romaguera, On completion of fuzzy metric spaces, Fuzzy Sets Syst.
130, 399–404, 2002.
- [15] V. Gregori and S. Romaguera, Fuzzy quasi-metric spaces, Appl. Gen. Topology 5,
129–136, 2004.
- [16] V. Gregori, S. Romaguera and A. Sapena, A characterization of bicpmpletable fuzzy
quasi-metric spaces, Fuzzy Sets Syst. 152, 395–402, 2005.
- [17] J.C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc. 12, 71–89, 1963.
- [18] E.P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer Academic Publishers,
Dordrecht, 2000.
- [19] I. Kramosil and J. Michálek, Fuzzy metric and statistical metric spaces, Kybernetica
11, 326–334, 1975.
- [20] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA. 28, 535–537, 1942.
- [21] I.L. Reilly, A generalized contraction principle, Bull. Austral. Math. Soc. 10, 359–363,
1974.
- [22] I.L. Reilly, I.L. Subrahmanyam and M.K. Vamanamurthy, Cauchy sequences in quasi-
pseudo-metric spaces, Monatsh. Math. 93, 127–140, 1982.
- [23] S. Romaguera, Left K-completeness in quasi-metric spaces, Math. Nachr. 157, 15–23,
1992.
- [24] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland series in prob-
ability and applied mathematics, Elsevier Science Publishing Co. Inc. 1983.
- [25] H. Sherwood, On the completion of probabilistic metric spaces, Z. Wahrsch. Verw.
Geb. 6, 62–64, 1966.
- [26] R.A. Stoltenberg, Some properties of quasi-uniform spaces, Proc. Lond. Math. Soc.
17, 226–240, 1967.
- [27] P.V. Subrahmanyam, Remarks on some fixed-point theorems related to Banach’s con-
traction principle, J. Math. Phys. Sci. 8, 455–457, 1974.
- [28] A. Wald, On a statistical generalization of metric spaces, Proc. Nat. Acad. Sci. USA.
29, 196–197, 1943.
- [29] W.A. Wilson, On quasi-metric spaces, Amer. J. Math. 53, 675–684, 1931.
- [30] J.R. Wu, X. Tang, Caristi’s fixed point theorem, Ekeland’s variational principle and
Takahashi’s maximization theorem in fuzzy quasi-metric spaces, Topology Appl. 302,
108701, 2021.
- [31] H. Yang and B. Pang, Fuzzy points based betweenness relations in L-convex spaces,
Filomat 35 (10), 3521–3532, 2021.
- [32] Y. Yue and J. Fang, Completeness in probabilistic quasi-uniform spaces, Fuzzy Sets
and Systems 370, 34–62, 2019.
- [33] L. Zhang and B. Pang, The category of residuated lattice valued filter spaces, Quaest.
Math. https://doi.org/10.2989/16073606.2021.1973140, 2021.
- [34] L. Zhang and B. Pang, Monoidal closedness of the category of $\top$-semiuniform conver-
gence spaces, Hacet. J. Math. Stat. 51 (5), 1348–1370, 2022.
- [35] F. Zhao and B. Pang, Equivalence among L-closure (interior) operators, L-closure
(interior) systems and L-enclosed (internal) relations, Filomat 36 (3), 979–1003,
2022.