##
**Another weak form of faint continuity.**
*(English)*
Zbl 0996.54016

Let \((X,\tau)\) be a topological space. A set \(A\) of \(X\) is said to be \(\alpha\)-open [O. Njåstad, Pac. J. Math. 15, 961-970 (1965; Zbl 0137.41903)] (resp. \(b\)-open [D. Andrijević, Mat. Vesn. 48, No. 1-2, 59-64 (1996; Zbl 0885.54002)] or sp-open [J. Dontchev, and M. Przemski, Acta Math. Hung. 71, No. 1-2, 109-120 (1996; Zbl 0852.54012)] or \(\gamma\)-open [A. A. El-Atik, A study of some types of mappings on topological spaces, MSc-Thesis, Tanta Univ., Egypt (1997)]) if \(A \subset\text{Int(Cl(Int}(A)))\), resp. \(A\subset\text{Cl(Int}(A))\cup \text{Int(Cl}(A))\). A subset \(A\) is said to be \(\theta\)-open [P. E. Long and L. L. Herrington, Kyungpook Math. J. 22, 7-14 (1982; Zbl 0486.54009)] if for each \(x\in A\) there exists an open set \(U\) such that \(x\in U\subset\text{Cl}(U)\subset A\). Long and Herrington defined a weak form of continuity by making use of \(\theta\)-open sets. T. Noiri and V. Popa [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 34(82), No. 3, 263-270 (1990; Zbl 0739.54003)] introduced and investigated three weaker forms of faint continuity which are called faint semicontinuity, faint precontinuity and faint \(\beta\)-continuity.

In this paper the author introduces and investigates other weak forms of faint continuity, namely faint \(\alpha\)-continuity and faint \(\gamma\)-continuity. A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be faintly \(\alpha\)-continuous (resp. faintly \(\gamma\)-continuous) if for each \(x\in X\) and each \(\theta\)-open set \(V\) of \(Y\) containing \(f(x)\) there exists an \(\alpha\)-open (resp. \(\gamma\)-open) set \(U\) containing \(x\) such that \(f(U)\subset V\).

Some characterizations and basic properties of these new types of functions are obtained. The relationships between these functions and several weak forms of faint continuity are investigated. Finally, the author mentions that the present paper may be relevant to fractal and Cantorian physics.

In this paper the author introduces and investigates other weak forms of faint continuity, namely faint \(\alpha\)-continuity and faint \(\gamma\)-continuity. A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be faintly \(\alpha\)-continuous (resp. faintly \(\gamma\)-continuous) if for each \(x\in X\) and each \(\theta\)-open set \(V\) of \(Y\) containing \(f(x)\) there exists an \(\alpha\)-open (resp. \(\gamma\)-open) set \(U\) containing \(x\) such that \(f(U)\subset V\).

Some characterizations and basic properties of these new types of functions are obtained. The relationships between these functions and several weak forms of faint continuity are investigated. Finally, the author mentions that the present paper may be relevant to fractal and Cantorian physics.

Reviewer: V.Popa (Bacau)

### MSC:

54C08 | Weak and generalized continuity |

### Keywords:

faint \(\beta\)-continuity; faint \(\alpha\)-continuity; faint \(\gamma\)-continuity; faint semicontinuity; faint precontinuity; faint continuity
PDF
BibTeX
XML
Cite

\textit{A. A. Nasef}, Chaos Solitons Fractals 12, No. 12, 2219--2225 (2001; Zbl 0996.54016)

Full Text:
DOI

### References:

[1] | AbdEl-Monsef, M.E; El-Deeb, S.N; Mahmoud, R.A, β-open sets and β-continuous mapping, Bull. fac. sci. assuit. univ., 12, 77-90, (1983) · Zbl 0577.54008 |

[2] | Andrijević, D, On the topology generated by preopen sets, Mate. bech., 39, 367-376, (1987) · Zbl 0646.54002 |

[3] | Andrijević, D, On b-open sets, Mat. bech., 48, 59-64, (1996) · Zbl 0885.54002 |

[4] | Dontchev, J; Przemski, M, On the various decompositions of continuous and some weakly continuous functions, Acta. math. hungar., 71, 1&2, 109-120, (1996) · Zbl 0852.54012 |

[5] | El-Atik AA. A study of some types of mappings on topological spaces. MSc Thesis, Tanta University, Egypt, 1997 |

[6] | Ganster, M, Preopen sets and resolvable spaces, Kyungpook math. J., 27, 135-143, (1987) · Zbl 0665.54001 |

[7] | Janković, D, θ-regular spaces, Int. J. math. sci., 8, 615-619, (1985) · Zbl 0577.54012 |

[8] | Levine, N, A decomposition of continuity in topological spaces, Amer. math. monthly, 63, 44-66, (1961) · Zbl 0100.18601 |

[9] | Levine, N, Semi-open sets and semi-continuity in topological spaces, Amer. math. monthly, 70, 36-41, (1963) · Zbl 0113.16304 |

[10] | Long, P.E; Herrington, L.L, The Tθ-topology and faintly continuous functions, Kyungpook math. J., 22, 7-14, (1982) · Zbl 0486.54009 |

[11] | Mashhour, A.S; Abd El-Monsef, M.E; El-Deeb, S.N, On precontinuous and weak precontinuous mappings, Proc. math. phys. soc. Egypt., 53, 47-53, (1982) · Zbl 0571.54011 |

[12] | Mashhour, A.S; Hasanein, I.A; El-Deeb, S.N, α-continuous and α-open mappings, Acta. math. hung., 41, 213-218, (1983) · Zbl 0534.54006 |

[13] | Njastad, O, On some classes of nearly open sets, Pacific J. math., 15, 961-970, (1965) · Zbl 0137.41903 |

[14] | Noiri, T, On δ-continuous functions, J. Korean math. soc., 16, 161-166, (1980) · Zbl 0435.54010 |

[15] | Noiri, T; Popa, V, Weak forms of faint continuity, Bull. math. soc. sci. math. roumanie, 34, 82, 270-363, (1990) · Zbl 0739.54003 |

[16] | Popa V, Stan G. On a decomposition of quasi-continuity in topological spaces. Stud Cerc Mat 1973;25:41-3 [in Romanian] |

[17] | Reilly, I.L; Vamanamurthy, M.K, On α-sets in topological spaces, Tamkang J. math., 16, 1, 7-11, (1985) · Zbl 0574.54009 |

[18] | Reilly, I.L; Vamanamurthy, M.K, Connectedness and strong semi-continuity, Časopisproestovani matematiky roč, 109, 261-265, (1984) · Zbl 0553.54005 |

[19] | Rose, D.A, Weak continuity and strongly closed sets, Int. J. math. sci., 7, 809-816, (1984) · Zbl 0592.54013 |

[20] | Singal, M.K; Singal, A.R, Almost continuous mappings, Yokohama math. J., 16, 63-73, (1968) · Zbl 0191.20802 |

[21] | Velicko, N.V, H-closed topological spaces, Amer. math. soc. transl., 78, 2, 103-118, (1968) · Zbl 0183.27302 |

[22] | Nottale, L, Fractal spacetime and microphysics, (1993), World Scientific Singapore |

[23] | El Naschie, M.S; Rossler, O.E; Prigogine, I, Quantum mechanics, diffusion and chaotic fractals, (1995), Elsevier-Pergamon Oxford · Zbl 0830.58001 |

[24] | El Naschie, M.S, Superstrings, knots and noncommutative geometry in \(E\^{}\{(∞)\}\) space, Int. J. theoret. phys., 37, 12, 2935-2951, (1998) · Zbl 0935.58005 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.