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Ordinary, Super and Hyper Relators Can Be Used To Treat the Various Generalized Open Sets in a Unified Way

In: Approximation and Computation in Science and Engineering

Author

Listed:
  • Themistocles M. Rassias

    (National Technical University of Athens)

  • Árpád Száz

    (University of Debrecen)

Abstract

If R $$\mathscr {R}$$ is a family of relations on X to Y , U $$\hskip 0.2 mm \mathscr {U}$$ is a family of relations on P ( X ) $$\mathscr {P}\hskip 0.2 mm(X)$$ to Y , and V $$\mathscr {V}$$ is a family of relations on P ( X ) $$\mathscr {P}\hskip 0.2 mm(X)$$ to P ( Y ) $$\mathscr {P}\hskip 0.2 mm(Y)$$ , then we say that R $$\mathscr {R}$$ is an ordinary relator, U $$\mathscr {U}$$ is a super relator, and V $$\mathscr {V}$$ is a hyper relator on X to Y . We show that the X = Y , U = { U } $$\mathscr {U}=\{\hskip 0.2 mm U\hskip 0.2 mm\}$$ and V = { V } $$\mathscr {V}=\{\hskip 0.2 mm V\hskip 0.2 mm\}$$ particular case of the non-conventional three relator space ( X , Y ) ( R , U , V ) $$\hskip 0.2 mm(\hskip 0.2 mm X\hskip 0.2 mm, \,Y\hskip 0.2 mm)(\hskip 0.2 mm\mathscr {R}\hskip 0.2 mm, \,\mathscr {U}\hskip 0.2 mm, \,\mathscr {V}\hskip 1.2 mm)$$ can be used to treat, in a unified way, the various generalized open sets studied by a great number of topologists.

Suggested Citation

  • Themistocles M. Rassias & Árpád Száz, 2022. "Ordinary, Super and Hyper Relators Can Be Used To Treat the Various Generalized Open Sets in a Unified Way," Springer Optimization and Its Applications, in: Nicholas J. Daras & Themistocles M. Rassias (ed.), Approximation and Computation in Science and Engineering, pages 709-782, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-84122-5_39
    DOI: 10.1007/978-3-030-84122-5_39
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