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**On \(\alpha\)-continuity in topological spaces.**
*(English)*
Zbl 0576.54014

In this paper \(\alpha\)-continuity, \(\alpha\)-open functions, and \(\alpha\)- closed functions, which were introduced in 1983 by A. Mashhour, I. Hasanein, and S. El-Deeb [Acta Math. Hung. 41, 213-218 (1983; Zbl 0534.54006)], and \(\alpha\)-irresolute functions, which were introduced in 1980 by S. Maheshwari and S. Thakur [Tamkang J. Math. 11, 209-214 (1981; Zbl 0485.54009)] are further investigated. The authors retopologize the domain space and/or range space to characterize \(\alpha\)-continuity using continuity, \(\alpha\)-open using open, \(\alpha\)- closed using closed, and \(\alpha\)-irresolute using continuity and these new characterizations are used to quickly establish many known properties of \(\alpha\)-continuous, \(\alpha\)-open, \(\alpha\)-closed, and \(\alpha\)- irresolute functions.

Reviewer: Ch.Dorsett

### MSC:

54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |

### Keywords:

\(\alpha \) -continuity; \(\alpha \) -open functions; \(\alpha \) -closed functions; \(\alpha \) -irresolute functions
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\textit{I. L. Reilly} and \textit{M. K. Vamanamurthy}, Acta Math. Hung. 45, 27--32 (1985; Zbl 0576.54014)

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### References:

[1] | N. Levine, Semi-open sets and semi-continuity in topological spaces,Amer. Math. Monthly,70 (1963), 36–41. · Zbl 0113.16304 |

[2] | S. N. Maheshwari and S. S. Thakur, On {\(\alpha\)}-irresolute mappings,Tamkang J. Math.,11 (1980), 209–214. |

[3] | A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings,Proc. Math. and Phys. Soc. Egypt,51 (1981). |

[4] | A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, {\(\alpha\)}-continuous and {\(\alpha\)}-open mappings,Acta Math. Hung.,41 (1983), 213–218. · Zbl 0534.54006 |

[5] | O. Njåstad, On some classes of nearly open sets,Pacific J. Math.,15 (1965), 961–970. · Zbl 0137.41903 |

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