In this paper, fuzzy hereditarily irresolvablity of a fuzzy topological space is established by means of closures of fuzzy sets. It is found that a fuzzy hyperconnected space is not a fuzzy perfectly disconnected space, while a fuzzy perfectly disconnected space is a fuzzy hereditarily irresolvable space. It is obtained that fuzzy hereditarily irresolvablity and fuzzy strongly irresolvablity of a fuzzy topological space are independent notions. Furthermore, it is established that the existence of fuzzy nowhere dense sets and fuzzy simply open sets ensures the non-hereditarily irresolvablity of a fuzzy topological space.

fuzzy dense set, fuzzy nowhere dense set, fuzzy simply open set, fuzzy hyperconnected space, fuzzy perfectly disconnected space, fuzzy submaximal space, fuzzy irresolvable space, fuzzy Baire space.

Received: November 23, 2021; Accepted: February 8, 2022; Published: March 2, 2022

How to cite this article: G. Thangaraj and L. Vikraman, On fuzzy hereditarily irresolvable spaces, Advances in Fuzzy Sets and Systems 27(1) (2022), 67-94. DOI: 10.17654/0973421X22004

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References:

[1] K. K. Azad, On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82 (1981), 14-32.
[2] G. Balasubramanian, Maximal fuzzy topologies, Kybernetika 31(5) (1995), 459-464.
[3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. 24 (1968), 182-190.
[4] J. Foran and P. Liebnitz, A characterization of almost resolvable spaces, Rend. Circ. Mat. Palermo (2) 40(1) (1991), 136-141.
[5] B. Ghosh, Fuzzy extremally disconnected spaces, Fuzzy Sets and Systems 46(2) (1992), 245-250.
[6] E. Hewitt, A problem of set theoretic topology, Duke Math. J. 10 (1943), 309-333.
[7] M. Katetov, On topological spaces containing no disjoint dense sets, Mat. Sb. 21 (1947), 3-12.
[8] Miguel Caldas, Govindappa Navalagi and Ratnesh Saraf, On fuzzy weakly semi-open functions, Proyecciones, Universidad Catolica del Norte, Antofagasta – Chile 21(1) (2002), 51-63.
[9] G. Thangaraj and G. Balasubramanian, On somewhat fuzzy continuous functions, J. Fuzzy Math. 11(2) (2003), 725-736.
[10] G. Thangaraj and G. Balasubramanian, On fuzzy resolvable and fuzzy irresolvable spaces, Fuzzy Sets, Rough Sets and Multivalued Operations and Appl. 1(2) (2009), 173-180.
[11] G. Thangaraj and S. Anjalmose, On fuzzy baire spaces, J. Fuzzy Math. 21(3) (2013), 667-676.
[12] G. Thangaraj and K. Dinakaran, On fuzzy simply continuous functions, J. Fuzzy Math. 25(1) (2017), 99-124.
[13] G. Thangaraj, B. Mathivathani and P. Sathya, On fuzzy resolvable sets and fuzzy resolvable functions, Adv. Fuzzy Math. 12(6) (2017), 1171-1181.
[14] G. Thangaraj and S. Lokeswari, Fuzzy irresolvable sets and fuzzy open hereditarily irresolvable spaces, Bull. Pure and Appl. Sci. 38(1) (2019), 369-377.
[15] G. Thangaraj and S. Lokeswari, Fuzzy strongly irresolvable spaces and fuzzy simply open sets, Adv. Fuzzy Sets and Sys. 27(1) (2022), 1-33.
[16] G. Thangaraj and S. Muruganantham, On fuzzy perfectly disconnected spaces, Inter. J. Adv. Math. 5 (2017), 12-21.
[17] R. H. Warren, Boundary of a fuzzy set, Indiana J. Math. 26 (1977), 191-197.
[18] L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353.