Research article

$ \delta $-connectedness modulo an ideal

  • Received: 23 March 2022 Revised: 17 July 2022 Accepted: 23 July 2022 Published: 05 August 2022
  • MSC : 03E15, 54D40, 54E05

  • The aim of this paper is to introduce the notion of $ \delta $-connectedness modulo an ideal in proximity spaces. A sufficient condition for a $ \delta $-connected modulo an ideal proximity space to be connected modulo an ideal is defined. The notion of $ \delta $-connectedness modulo a proximal property is defined, and several results for $ \delta $-connectedness modulo compactness and modulo pseudocompactness are obtained. $ \delta $-perfect map for proximity spaces is defined and it is shown that the class of $ \delta $-perfect maps is properly contained in the class of perfect maps, and some results about $ \delta $-perfect maps are substantiated.

    Citation: Beenu Singh, Davinder Singh. $ \delta $-connectedness modulo an ideal[J]. AIMS Mathematics, 2022, 7(10): 17954-17966. doi: 10.3934/math.2022989

    Related Papers:

  • The aim of this paper is to introduce the notion of $ \delta $-connectedness modulo an ideal in proximity spaces. A sufficient condition for a $ \delta $-connected modulo an ideal proximity space to be connected modulo an ideal is defined. The notion of $ \delta $-connectedness modulo a proximal property is defined, and several results for $ \delta $-connectedness modulo compactness and modulo pseudocompactness are obtained. $ \delta $-perfect map for proximity spaces is defined and it is shown that the class of $ \delta $-perfect maps is properly contained in the class of perfect maps, and some results about $ \delta $-perfect maps are substantiated.



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    [1] R. Dimitrijević, Lj. Kočinac, On connectedness of proximity spaces, Mat. Vesn., 39 (1987), 27–35.
    [2] V. A. Efremovic, The geometry of proximity, Mat. Sb., 31 (1952), 189–200.
    [3] E. Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc., 64 (1948), 45–99. https://doi.org/10.1090/S0002-9947-1948-0026239-9 doi: 10.1090/S0002-9947-1948-0026239-9
    [4] M. R. Koushesh, Connectedness modulo a topological property, Topol. Appl., 159 (2012), 3417–3425. https://doi.org/10.1016/j.topol.2012.08.001 doi: 10.1016/j.topol.2012.08.001
    [5] M. R. Koushesh, Connectedness modulo an ideal, Topol. Appl., 214 (2016), 150–179. https://doi.org/10.1016/j.topol.2016.10.009 doi: 10.1016/j.topol.2016.10.009
    [6] K. Kuratowski, Topology, Elsevier, 1966.
    [7] S. G. Mrówka, W. J. Pervin, On uniform connectedness, P. Am. Math. Soc., 15 (1964), 446–449. https://doi.org/10.2307/2034521 doi: 10.2307/2034521
    [8] J. Munkres, Topology, Prentice Hall, 2000.
    [9] S. Naimpally, Proximity approach to problems in topology and analysis, München: Oldenbourg Wissenschaftsverlag, 2010. https://doi.org/10.1524/9783486598605
    [10] S. Naimpally, B. D. Warrack, Proximity spaces, 1970.
    [11] B. Singh, D. Singh, Sum connectedness in proximity spaces, Appl. Gen. Topol., 22 (2021), 345–354. https://doi.org/10.4995/agt.2021.14809 doi: 10.4995/agt.2021.14809
    [12] B. Singh, D. Singh, $S$-$ \delta $-connectedness in $S$-proximity spaces, Commun. Fac. Sci. Univ., 70 (2021), 600–611. https://doi.org/10.31801/cfsuasmas.792265 doi: 10.31801/cfsuasmas.792265
    [13] B. Singh, D. Singh, Connectedness in ideal proximity spaces, Honam Math. J., 43 (2021), 123–129. https://doi.org/10.5831/HMJ.2021.43.1.123 doi: 10.5831/HMJ.2021.43.1.123
    [14] Y. M. Smirnov, On completeness of proximity spaces I, Amer. Math. Soc. Trans., 38 (1964), 37–73.
    [15] Y. M. Smirnov, On the completeness of proximity spaces, Tr. Mosk. Mat. Obs., 3 (1954), 271–306.
    [16] R. Vaidyanathaswamy, The localisation theory in set topology, Proc. Indian Acad. Sci., 20 (1944), 51–61. https://doi.org/10.1007/BF03048958 doi: 10.1007/BF03048958
    [17] S. Willard, General topology, Addison-Wesley, 1970.
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