Weak neighborhoods and Michael-Nagami's question (Q2705851)

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 compact-covering mapping if each compact subset of \(Y\) is the image of some compact subset of \(X\) under \(f\). A study of 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 0245.54023)] posed the following question. If a Hausdorff space \(X\) is a quotient \(s\)-image of a metric space, must \(X\) also be a compact-covering quotient \(s\)-image of a (possibly different) metric space? \textit{E. Michael} posed the question again in [Open problems in topology, (1990; Zbl 0718.54001) 273-278]. \textit{G. Gruenhage}, \textit{E. Michael} and \textit{Y. Tanaka} in [Pac. J. Math. 113, 303-332 (1984; Zbl 0561.54016)] proved that if a Hausdorff space \(X\) is a quotient \(s\)-image of a metric space, then \(X\) also is a sequence-covering quotient \(s\)-image of a (possibly different) metric space. \textit{Shou Lin} and \textit{Chuan Liu} in [Topology Appl. 74, No. 1-3, 51-60 (1996; Zbl 0869.54036)] proved that a Hausdorff sequential space with a point-countable cs-network is a compact-covering quotient \(s\)-image of a metric space. In this paper the author constructs a Hausdorff space \(X\) such that \(X\) is a quotient \(s\)-image of a locally separable metric space and is no compact-covering quotient \(s\)-image of any metric space, which negatively answers Michael-Nagami's question.