Properties of quasi-precontinuous functions
A subset \(A\) of a topological space \(X\) is called preopen if \(A\subset \text{Int(Cl} (A))\). A function \(f:X\to Y\) is called quasi-precontinuous [the authors, Quasi-pre-continuous functions, J. Indian Acad. Math. 14, No. 2, 115-126 (1992)] if for each \(x\in X\) and each open set \(V\) containing \(f(x)\) there exists a preopen set \(U\) containing \(x\) such that \(f(U) \subset \text{Cl} (V)\). Recently, the reviewer and \textit{T. Noiri} [Demonstr. Math. 25, No. 1/2, 241-251 (1992; Zbl 0789.54014), Theorem 3.1] showed that quasi-precontinuity is equivalent to almost weak continuity due to \textit{D. S. Janković} [Int. J. Math. Math. Sci. 8, 615-619 (1985; Zbl 0577.54012)]. The authors study some properties of quasi precontinuous functions and the reviewer emphasizes that they are also properties of weakly almost continuous functions. But almost all of them are already known, as they appeared either in the above mentioned paper of Popa and Noiri or in [\textit{T. Noiri}, Int. J. Math. Math. Sci. 10, 97-111 (1987; Zbl 0617.54008)] or in [\textit{T. Noiri} and \textit{V. Popa}, Demonstr. Math. 26, No. 2, 363-380 (1993; Zbl 0859.54007)]. So Theorem 3.1 is a mixture of some sentences from Theorem 3.1 of [Noiri, loc. cit.] and Theorem 3.1 of [Popa and Noiri, [loc. cit.]. Theorem 4.1 is Theorem 3.7 from [Popa and Noiri, loc. cit.]. Theorem 4.2 is Theorem 6.3.5 from [Noiri, loc. cit.]. Theorem 4.7 is a particular case of Theorem 7.2 from [Noiri and Popa, loc. cit.]. Theorem 4.8 is a particular case of Theorem 7.1 from [Noiri and Popa, loc. cit.]. Theorem 5.5 is Theorem 5.6 from [Popa and Noiri, [loc. cit.]. Corollary 5.7 is Theorem 5.1 from [Popa and Noiri, loc. cit.]. Theorem 6.1 is a consequence of Theorem 6.1 from [Noiri and Popa, loc. cit.]. Theorem 6.2 is a particular case of Corollary 5.3 from [Noiri and Popa, loc. cit.]. Theorem 6.3 is a particular case of Theorems 4.6 and 4.8 from [Noiri and Popa, loc. cit.]. From the ``References'' results that the authors know at least the papers [Noiri, loc. cit.] and [Popa and Noiri, loc. cit.].
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