A question paralleling to Michael-Nagami problem (Q2703089)

Let \(f:X\to Y\) be a continuous and onto mapping. \(f\) is an \(s\)-mapping if each \(f^{-1}(y)\) is separable in \(X\); \(f\) is a \(cs\)-mapping if \(f^{-1}(C)\) is separable in \(X\) for each compact subset \(C\) of \(Y\). A study of the quotient \(s\)-image of a metric space is an interesting question. \textit{E. Michael} and \textit{K. Nagami} in [Proc. Am. Math. Soc. 37, 260-266 (1973; Zbl 0228.54008)] posed the following question: Suppose \(X\) is a quotient \(s\)-image of a metric space, is \(X\) a compact-covering and \(s\)-image of a metric space? NEWLINENEWLINENEWLINERecently, \textit{Huaipeng Chen} [Weak neighborhoods and Michael-Nagami's question, Houston J. Math. 25, No. 2, 297-309 (1999)] negatively answers Michael-Nagami's question. In this paper the author obtains a characterization of compact-covering \(cs\)-image of a metric space, and poses the following question: Suppose \(X\) is a quotient \(cs\)-image of a metric space, is \(X\) a compact-covering and \(cs\)-image of a metric space?NEWLINENEWLINENEWLINEThe reviewer does not know whether the counterexample in Huaipeng's paper is a counter-example to the question above.