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On the lattice of principal generalized topologies

Periodica Mathematica Hungarica volume 78, pages 79–87 (2019)Cite this article

Abstract

In this paper we investigate the structure of the complete lattice of principal generalized topologies, employing the notion of ultratopology. On any partially ordered set we introduce a generalized topology. The existence of an anti-isomorphism between principal generalized topologies and preorder relations on a set is proposed. After determining the very basic topological properties therein, we will show that each generalized topology has a lattice complement in principal generalized topologies.

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References

  1. R. Baskaran, M. Murugalingam, D. Sivaraj, Lattice of generalized topology. Acta Math. Hung. 133(4), 365–371 (2011)

    Article  MATH  Google Scholar 

  2. Á. Császár, Foundations of General Topology (Pergamon Press, London, 1963)

    MATH  Google Scholar 

  3. Á. Császár, Generalized open sets. Acta Math. Hung. 75, 65–87 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Á. Császár, Generalized topology, generalized continuity. Acta Math. Hung. 96, 351–357 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Á. Császár, Ultratopologies generated by generalized topologies. Acta Math. Hung. 110(1–2), 153–157 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Á. Császár, On generalized neighborhood systems. Acta Math. Hung. 121(4), 395–400 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Gerlits, G. Sági, Ultratopologies. Math. Logic Q. 50(6), 603–612 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Gaifman, Remarks on complementation in the lattice of all topologies. Can. J. Math. 18, 83–88 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  9. D.C. Kent, W.J. Min, Neighborhood spaces. Int. J. Math. Sci. 32, 387–399 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. R.E. Larson, S.K. Andima, The lattice of topologies: a survey. Rockey Mountain J. Math. 5(2), 177–198 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Lorrian, Notes on topological spaces with minimal neighborhoods. Am. Math. Monthly 76, 616–627 (1969)

    Article  Google Scholar 

  12. W.K. Min, On ascending generalized neighborhood spaces. Acta. Sci. Math. Hung. 127, 391–396 (2010)

    Article  MATH  Google Scholar 

  13. G. Sági, Ultraproducts and higher order formulas. Math. Logic Q. 48(2), 261–275 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Sági, S. Shelah, On Topological Properties of Ultraproducts of Finite Sets. Math. Logic Q. 51(3), 254–257 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Á. Száz, Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities. Filomat 21, 87–97 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. A.K. Steiner, The lattice of topologies: structure and complementation. Trans. Am. Math. Soc. 122, 379–397 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  17. A.K. Steiner, Complementation in the lattice of \(T_1\)-topologies. Proc. Am. Math. Soc. 17, 884–886 (1966)

    MATH  Google Scholar 

  18. A.K. Steiner, E.F. Steiner, Topologies with \(T_1\)-complements. Fund. Math. 61, 23–28 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. W.J. Thron, Topological Structures (Holt, Rienehart and Winston, New York, 1966)

    MATH  Google Scholar 

  20. A.C.M. Van Rooji, The lattice of all topologies is complemented. Can. J. Math. 20, 805–807 (1968)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank to the referee for carefully reading the paper and for giving useful comments which will help to improve it.

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  1. Department of Mathematics, Tafresh University, Tafresh, 39518-79611, Iran

    Hassan Arianpoor

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  1. Hassan Arianpoor
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Correspondence to Hassan Arianpoor.

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Arianpoor, H. On the lattice of principal generalized topologies. Period Math Hung 78, 79–87 (2019). https://doi.org/10.1007/s10998-018-0266-8

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  • DOI: https://doi.org/10.1007/s10998-018-0266-8

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