Partition-complete spaces are preserved by tri-quotient maps
Partition-completeness, introduced in \textit{E. Michael} [Proc. Am. Math. Soc. 96, 513-522 (1986; Zbl 0593.54028)] and studied further in [the reviewer, Complete exhaustive sieves, to appear] and \textit{R. Telgársky} and the reviewer [Complete exhaustive sieves and games, Proc. Am. Math. Soc. 102, No. 3, 737-744 (1988; Zbl 0643.54028)], generalizes both Čech-completeness and sieve-completeness. It was shown in the just cited paper that partition-completeness is preserved by both perfect maps and open maps. These two kinds of maps are generalized by tri-quotient maps which were introduced and studied in [\textit{E. Michael}, Ill. J. Math. 21, 716-733 (1977; Zbl 0386.54007)]. In the present article the author proves that tri-quotient maps preserve partition-completeness. The proof involves more than a straightforward generalization of the proof for perfect maps and is accomplished via several lemmas on partition- completeness and tri-quotient maps. The author asks whether every tri- quotient map \(f:X \to Y\) on a regular partition-complete space \(X\) to a paracompact space \(Y\) is inductively perfect, which, if true, would generalize the corresponding theorem when \(X\) is sieve-complete, proved in [Michael (1977) loc. cit.]. The answer is no as shown by \textit{W. Just} and the reviewer in example 5.1 in [Some conditions under which tri- quotient or compact-covering maps are inductively perfect, Topology Appl. 55, No. 3, 289-305 (1994; see the review below)].
scientific article; zbMATH DE number 123579