$ \diamond $-hyperconnected spaces
DOI:
https://doi.org/10.5269/bspm.76197Abstract
The purpose of this paper is to introduce and study the concept of $ \diamond $-hyperconnectedness as a generalization of hyperconnectedness in ideal topological spaces. Several characterizations of it are established and it is shown that hyperconnectedness and $ \diamond $-hyperconnectedness coincide in case of trivial and codense ideal.
References
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[2] N. Bourbaki, General Topology, Springer-Verlag Berlin, (1998).
[3] E. Ekici and T. Noiri, ∗-hyperconnected ideal topological spaces, Analele Stiintifice Ale Universitatii Al. I. Cuza Din Iasi (S.N.) Matematica Tomul LVIII, f. 1, 121−129, (2012).
[4] B. Garai and C. Bandyopadhyay, Nowhere dense sets and hyperconnected s-topological spaces, Bull. Calcutta Math. Soc. 92, 55−58, (2000).
[5] D. Jankovic and T. R. Hamlett, New Topologies from old via Ideals, Amer. Math. Monthly 97, 295−310, (1990).
[6] J.L. Kelley, General Topology, Springer, (1991).
[7] K. Kuratowski, Topology Vol.1, New York Academic Press, (1966).
[8] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70, 36−41, (1963).
[9] N. Levine, Dense topologies, Amer. Math. Monthly 75, 847−852, (1968).
[10] S.N. Maheshwari and U. Tapi, Connectedness of a stronger type in topological spaces, Nanta Math. 12, 102−109, (1979).
[11] P.M. Mathew, On hyperconnected spaces, Indian J. Pure Appl. Math. 19, 1180−1184, (1988).
[12] N.R. Pachón Rubiano, A generalization of connectedness via ideals, Armen. J. Math. 14, Paper No. 7, 18 pp, (2022).
[13] V. Pipitone and G. Russo, Spazi semiconnessi e spazi semiaperti, Rend. Circ. Mat. Palermo 24, 273−285, (1975).
[14] B. Singh and D. Singh, Connectedness in ideal proximity spaces, Honam Math. J. 43(1), 123−129, (2021).
[15] B. Singh and D. Singh, δ-connectedness modulo an ideal, AIMS Math. 7(10),17954−17966, (2022).
[16] L.A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, Holt, Rinehart and Winster, New York, (1970).
[17] T. Thompson, Characterizations of irreducible spaces, Kyungpook Math J. 21, 191−194, (1981).
[18] B.K. Tyagi, M. Bhardwaj and S. Singh, Hyperconnectedness modulo an ideal, Filomat 32(18), 6375−6386, (2018).
[19] R. Vaidyanathaswamy, The localisation theory in set topology, Proc. Indian Acad. Sci. 20, 51−61, (1944)
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Published
2025-09-22
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How to Cite
Singh, B., & Deep, A. (2025). $ \diamond $-hyperconnected spaces. Boletim Da Sociedade Paranaense De Matemática, 43, 1-7. https://doi.org/10.5269/bspm.76197
