Baire category theorem for the symmetric topology of sublinear quasi-metrics in locally convex cones

Document Type : Original Paper


Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran


In this paper, we investigate the completeness of cones in the symmetric topology ‎induced ‎the ‎sublinear ‎quasi-metrics‎ and prove that in the symmetric complete cone the symmetric neighborhoods are of the second category. Then we present an extension of the Baire category theorem for the symmetric topology of locally convex quasi-metric cones.


Main Subjects

[1] Azizi Mayvan A. and Motallebi M.R., Ekeland’s type variational principle for locally convex conevalued
functions, J. Fixed Point Theory Appl., 23(4) (2021).
[2] Azizi Mayvan A. and Motallebi M.R., Pointwise well-posedness and scalarization of optimization
problems for locally convex cone-valued functions, Filomat, 34(5) (2020), 1571–1579.
[3] Farajzadeh A.P., On the scalarization method in cone metric spaces, Positivity, 18(4) (2014), 703-708.
[4] Keimel K. and Roth W., Ordered cones and approximation, Lecture Notes in Mathematics, vol. 1517,
Springer Verlag, Heidelberg-Berlin-New York, 1992.
[5] Megginson R.E., An introduction to Banach space theory, Springer-Verlag, New York, 1998.
[6] Motallebi M.R., Completeness on locally convex cones, C. R. Math. Acad. Sci. Paris 352(10) (2014),
[7] Motallebi M.R., On weak completeness of products and direct sums in locally convex cones, Period.
Math. Hung., 75(2) (2017), 322-329.
[8] Motallebi M.R., Weak compactness of direct sums in locally convex cones, Stud. Sci. Math. Hung.,
55(4) (2018), 487-497.
[9] Motallebi M.R. and Saiflu H., Products and direct sums in locally convex cones, Can. Math. Bull.,
55(4) (2012), 783-798.
[10] Rudin W., Real and complex analysis, McGraw-Hill Inc., New York, 1974.
[11] Tavakoli M, Farajzadeh A.P, Abdeljawad T. and Suantai S., Some notes on cone metric spaces, Thai
J. Math., 16(1)(2018), 229-242.
[12] Yousefi Z. and Motallebi M.R., On sublinear quasi-metrics and neighborhoods in locally convex
cones, Filomat, 36(3) (2022), 721-728.
[13] Zangenehmehr P, Farajzadeh A.P. and Vaezpour S.M., On fixed point theory for generalized contractions
in cone metric spaces via scalarizing, Chiang Mai J. Sci., 42(4) (2015), 1038-1043.
Volume 12, Issue 2
June 2022
Pages 201-211
  • Receive Date: 01 March 2022
  • Revise Date: 17 May 2022
  • Accept Date: 17 May 2022
  • First Publish Date: 13 June 2022