Some conditions under which tri-quotient or compact-covering maps are inductively perfect
The principal results are the following two theorems, which answer Questions 1.1 and 1.2 in the reviewers note [Some problems, in: \textit{J. van Mill} and \textit{G. M. Reed} (eds.) Open problem in topology (1990; Zbl 0718.54001); 273-278]. (A) (Theorem 2.4): If \(f:X \to Y\) is a countable-compact-covering map with compact fibers from a separable metrizable space \(X\) onto a metrizable space \(Y\), then \(f\) is inductively perfect. (B) (From Theorems 3.1 and 3.2): If \(f:X \to Y\) is a compact- covering map from a separable metrizable space \(X\) onto a countable metrizable space \(Y\), then \(f\) is inductively perfect. [Note: (1) A map \(f:X \to Y\) is called (countable-) compact-covering if every (countable and) compact \(K \subset Y\) is the image of some compact \(C \subset X\); it is called inductively perfect if there is an \(X' \subset X\) such that \(f(X')=Y\) and \(f | X'\) is perfect. (2) The proof of Theorem 2.4 uses a result by \textit{H. M. Cho} and \textit{W. Just} [to appear in Topology Appl.]. There are also two related results: (C) (Theorem 4.0): A result of \textit{A. V. Ostrovskij} [Topology Appl. 23, 25-28 (1986; Zbl 0591.54010)] is generalized by weakening ``\(G_ \delta\)-set'' to ``\(W_ \delta\)-set''. (D) (Example 5.1). There exists an open map with compact fibers from a Tykhonov space \(X\) with one nonisolated point onto a Hausdorff space \(Y\) which is not compact-covering. (This implies that, in Theorem 6.6 of the reviewer's paper [Ill. J. Math. 21, 716-733 (1977; Zbl 0386.54007)], ``sieve-complete'' cannot be weakened to ``partition- complete'').
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