\(\mathcal{I}^{\mathcal{K}}\)-SEQUENTIAL TOPOLOGY

H. S. Behmanush     (Mersin University, Science Faculty, Department of Mathematics, 33100 Yenişehir, Mersin, Turkey)
M. Küçükaslan     (Mersin University, Science Faculty, Department of Mathematics, 33100 Yenişehir, Mersin, Turkey)

Abstract


In the literature, \(\mathcal{I}\)-convergence (or convergence in \(\mathcal{I}\)) was first introduced in [11]. Later related notions of \(\mathcal{I}\)-sequential topological space and \(\mathcal{I}^*\)-sequential topological space were introduced and studied. From the definitions it is clear that \(\mathcal{I}^*\)-sequential topological space is larger(finer) than \(\mathcal{I}\)-sequential topological space. This rises a question: is there any topology (different from discrete topology) on the topological space \(\mathcal{X}\) which is finer than \(\mathcal{I}^*\)-topological space? In this paper, we tried to find the answer to the question. We define \(\mathcal{I}^{\mathcal{K}}\)-sequential topology for any ideals \(\mathcal{I}\), \(\mathcal{K}\) and study main properties of it. First of all, some fundamental results about \(\mathcal{I}^{\mathcal{K}}\)-convergence of a sequence in a topological space \((\mathcal{X} ,\mathcal{T})\) are derived. After that, \(\mathcal{I}^{\mathcal{K}}\)-continuity and the subspace of the \(\mathcal{I}^{\mathcal{K}}\)-sequential topological space are investigated.


Keywords


Ideal convergence, \(\mathcal{I}^{\mathcal{K}}\)-convergence, Sequential topology, \(\mathcal{I}^{\mathcal{K}}\)-sequential topology

Full Text:

PDF

References


  1. Blali A., El Amrani A., Hasani R.A., Razouki A. On the uniqueness of \(\mathcal{I}\)-limits of sequences. Siberian Electron. Math. Rep., 2021. Vol. 8, No. 2. P. 744–757. DOI: 10.33048/semi.2021.18.055
  2. Banerjee A.K., Paul M. Strong \(\mathcal{I}^{\mathcal{K}}\)-convergence in probabilistic metric spaces. Iranian J. Math. Sci. Inform., 2022. Vol. 17, No. 2. P. 273–288. DOI: 10.52547/ijmsi.17.2.273
  3. Banerjee A.K., Paul M. A note on \(I^{K}\) and \(I^{K^{*}}\)-Convergence in Topological Spaces. 2018. 10 p. arXiv:1807.11772v1 [math.GN]
  4. Das P. Some further results on ideal convergence in topological spaces. Topology Appl., 2012. Vol. 159, No. 10–11. P. 2621–2626. DOI: 10.1016/j.topol.2012.04.007
  5. Das P., Sleziak M., Toma V. \(\mathcal{I}^{\mathcal{K}}\)-Cauchy functions. Topology Appl., 2014. Vol. 173. P. 9–27. DOI: 10.1016/j.topol.2014.05.008
  6. Das P., Sengupta S., Šupina J. \(\mathcal{I}^{\mathcal{K}}\)-convergence of sequence of functions. Math. Slovaca, 2019. Vol. 69, No. 5. P. 1137–1148. DOI: 10.1515/ms-2017-0296
  7. Fast H. Sur la convergence statistique. Colloq. Math., 1951. Vol. 2, No. 3–4. P. 241–244. URL: https://eudml.org/doc/209960 (in French)
  8. Georgiou D., Iliadis S., Megaritis A., Prinos G. Ideal-convergence classes. Topology Appl., 2017. Vol. 222. P. 217–226. DOI: 10.1016/j.topol.2017.02.045
  9. Jasinski J., Recław I. Ideal convergence of continuous functions. Topology Appl., 2006. Vol. 153, No. 18. P. 3511–3518. DOI: 10.1016/j.topol.2006.03.007
  10. Kostyrko P., Mačaj M., Šalát T., Sleziak M. \(\mathcal{I}\)-convergence and extremal \(\mathcal{I}\)-limit points. Math. Slovaca,  2005. Vol. 55, No. 4. P. 443–464.
  11. Kostyrko P., Salát T., Wilczyńki W. \(\mathcal{I}\)-convergence. Real Anal. Exchange, 2000/2001. Vol. 26, No. 2. P. 669–685.
  12. Lahiri B.K., Das P. \(I\) and \(I^∗\)-convergence in topological space. Math. Bohem., 2005. Vol. 130, No. 2. P. 153–160. DOI: 10.21136/MB.2005.134133
  13. Mačaj M., Sleziak M. \(\mathcal{I}^{\mathcal{K}}\)-convergence. Real Anal. Exchange, 2010–2011. Vol. 36, No. 1. P. 177–194.
  14. Mursaleen M., Debnath S., Rakshit D. \(I\)-statistical limit superior and \(I\)-statistical limit inferior. Filomat, 2017. Vol. 31, No. 7. P. 2103–2108. DOI: 10.2298/FIL1707103M
  15. Öztürk F., Şençimen C., Pehlivan S. Strong \(\Gamma\)-ideal convergence in a probabilistic normed space. Topology Appl., 2016. Vol. 201. P. 171–180. DOI: 10.1016/j.topol.2015.12.035
  16. Pal S.K. \(\mathcal{I}\)-sequential topological space. Appl. Math. E-Notes, 2014. Vol. 14. P. 236–241.
  17. Renukadevi V., Prakash B. \(\mathcal{I}\): Fréchet-Urysohn spaces. Math. Morav., 2016. Vol. 20, No. 2. P. 87–97. DOI: 10.5937/MatMor1602087R
  18. Sabor Behmanush H., Küçükaslan M. Some Properties of \(\mathcal{I}^*\)-Sequential Topological Space. 2023. 13 p. arXiv:2305.19647 [math.GN]
  19. Salát T., Tripathy B.C., Ziman M. On some properties of \(\mathcal{I}\)-convergence. Tatra Mt. Math. Publ., 2004. Vol. 28, No. 2. P. 279–286.
  20. Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math., 1951. Vol. 2. P. 73–74. (in French)
  21. Zhou X., Liu L., Lin S. On topological space defined by \(\mathcal{I}\)-convergence. Bull. Iran. Math. Soc., 2020. Vol. 46, No. 3. P. 675–692. DOI: 10.1007/s41980-019-00284-6




DOI: http://dx.doi.org/10.15826/umj.2023.2.004

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.